1. Introduction

The cooling-to-space approximation is a venerable tool of radiative transfer. Formulated over 50 years ago (M. Zagoni 2016, unpublished manuscript; Green 1967; Rodgers and Walshaw 1966), it gives a simplified description of radiative cooling suitable for textbooks (Wallace and Hobbs 2006; Petty 2006; Thomas and Stamnes 2002), heuristics and idealized modeling (Jeevanjee and Fueglistaler 2020; Jeevanjee and Romps 2018), and has served in the past as a basis for comprehensive radiation schemes (Eymet et al. 2004; Joseph et al. 1976; Fels and Schwarzkopf 1975; Rodgers and Walshaw 1966). Its content is simply that radiative cooling in a given layer can be approximated as that layer’s emission or cooling to space (CTS), as radiative exchange between atmospheric layers can be neglected. This claim is quite intuitive, as the exchange terms are sourced by the temperature difference between layers, whereas the CTS term is sourced by the absolute temperature of a layer. Furthermore, exchange with cooler layers above offsets exchange with warmer layers below. This “double cancellation” in the sum of exchange terms should render that sum negligible.

This logic is plausible, and the CTS approximation indeed seems to hold quite well for terrestrial atmospheric profiles (e.g., Clough et al. 1992; Rodgers and Walshaw 1966; Fig. 6 below). However, to our knowledge a formal justification has never been given. We thus have no precise understanding of why the CTS approximation works in Earth’s atmosphere, or under what conditions it might fail (on Earth or elsewhere). Furthermore, it is clear that the CTS approximation does fail in some cases, such as the textbook case of a gray gas in pure radiative equilibrium (PRE; e.g., Pierrehumbert 2010). This state has zero radiative cooling by definition, and thus the CTS term must be entirely canceled by the exchange terms. We are thus led to the following questions:

1. Under what conditions does the CTS approximation hold?

2. How do these conditions break down (as they must in PRE), and how can this be reconciled with the double cancellation argument given above?

The goal of this paper is to shed light on these questions. A key ingredient in our analysis will be a refinement of the canonical decomposition of radiative flux divergence given by Green (1967) into a new decomposition which naturally captures the double cancellation described above, and which also isolates the contributions which do not cancel (section 2). We apply this framework to gray PRE, to understand how the CTS approximation can break down (section 3). In sections 4 and 5 we then turn to gray gas radiative–convective equilibrium (RCE), to understand how the CTS approximation emerges, along with its concomitant “τ = 1 law” (Huang and Shahabadi 2014). Finally in section 6 we consider cooling from realistic greenhouse gases, using the line-by-line Reference Forward Model (Dudhia 2017). Along the way we construct validity criteria for the CTS approximation capable of explaining its breakdown for the PRE state as well as its success for H₂O in RCE, and which also correctly predicts that the CTS approximation should hold only marginally for CO₂ in RCE. We also consider the impact of the choice of vertical coordinate on the CTS approximation and the τ = 1 law. We do not consider radiative heating by solar absorption, though this is formally very similar to the CTS term (e.g., Jeevanjee and Romps 2018), so any results concerning the CTS term also apply to solar absorption.

2. A new decomposition of radiative flux divergence

a. Derivation

We begin by constructing a new decomposition of radiative flux divergence. For clarity and simplicity we do this first for a two-stream gray gas using optical depth τ as the vertical coordinate, extending our analysis later to realistic gases and the more conventional pressure coordinate.

The new decomposition we pursue is in some sense a refinement of the standard decomposition of radiative flux divergence found in textbooks, which says that radiative cooling in a given atmospheric layer can be decomposed into that layer’s cooling to space, as well as its radiative exchange with layers above and below as well as the ground. Formally this can be expressed as (e.g., Petty 2006; Thomas and Stamnes 2002; Green 1967; Rodgers and Walshaw 1966):

${\displaystyle \frac{dF}{d\tau }}=\left[B_s-B\left(\tau \right)\right]\hspace{0.17em}\exp \left[-\left({\tau }_s-\tau \right)\right]\left(\text{GX}\right)$(1a)

$-\hspace{0.17em}B\left(\tau \right)\hspace{0.17em}\exp \left(-\tau \right)\left(\text{CTS}\right)$(1b)

$+{{\displaystyle \int }}_{\tau }^{{\tau }_s}\left[B\left({\tau }^{\prime }\right)-B\left(\tau \right)\right]\hspace{0.17em}\exp \left[-\left({\tau }^{\prime }-\tau \right)\right]\hspace{0.17em}d{\tau }^{\prime }\left({\text{EX}}_{\text{below}}\right)$(1c)

$+{{\displaystyle \int }}_0^{\tau }\left[B\left({\tau }^{\prime }\right)-B\left(\tau \right)\right]\hspace{0.17em}\exp \left[-\left(\tau -{\tau }^{\prime }\right)\right]\hspace{0.17em}d{\tau }^{\prime }\left({\text{EX}}_{\text{above}}\right).$(1d)

Here F is the net upward flux (W m⁻²), τ_s is the surface optical depth, and B_s is the value of the source function at the surface, which may be discontinuous from the source function B(τ_s) of the atmosphere at the surface. The GX term in (1a) is the “ground exchange” term, representing exchange between level τ and the surface. The CTS term in Eq. (1b) is the product of the source function and the transmissivity e^−τ, and thus represents the cooling to space, which is also the contribution of level τ to the outgoing longwave radiation. The CTS approximation is then just the claim that this term dominates Eq. (1), that is, that

${\partial }_{\tau }F\approx -B\left(\tau \right)e^{-\tau }\left(\text{CTS approximation}\right).$(2)

The EX_below and EX_above terms in Eqs. (1c) and (1d) represent radiative exchange with layers below and above. As mentioned above, these terms are generated by temperature differences, unlike the CTS term, and since temperatures change monotonically over large swaths of atmosphere (the troposphere, stratosphere, etc.), EX_below and EX_above typically have opposite signs and thus offset each other. A key point, however, is that the degree to which they cancel is in part tied to their respective ranges of integration: if τ_s − τ is not comparable to τ, that is, if the emissivity above level τ is not comparable to that below τ, then the integrals in Eqs. (1c) and (1d) will not be comparable, inhibiting cancellation. This possibility is key for understanding how the CTS approximation can break down.

To separate out the parts of EX_below and EX_above which might cancel each other, we first change the dummy integration variable in Eqs. (1c) and (1d) to measure the optical distance from level τ, that is, we set x ≡ τ′ − τ in (1c) and x ≡ τ − τ′ in Eq. (1d). This yields

$\begin{array}{l}{\text{EX}}_{\text{below}}={\displaystyle {\int }_0^{{\tau }_s-\tau }\left[B\left(\tau +x\right)-B\left(\tau \right)\right]e^{-x}\hspace{0.17em}dx},\\ {\text{EX}}_{\text{above}}=-{\displaystyle {\int }_0^{\tau }\left[B\left(\tau \right)-B\left(\tau -x\right)\right]e^{-x}\hspace{0.17em}dx}.\end{array}$

The EX_below and EX_above integrals now look similar, but with potentially different limits of integration. Cancellation will most naturally occur between those parts of the integral with the same range of integration, so we combine them. To do this we first consider the case where τ < τ_s/2 (i.e., the layer below is optically deeper than that above, Fig. 1a). In this case we split the EX_below integral into an integral over the x interval (0, τ) (same as the range for the EX_above integral), and an integral over the x interval (τ, τ_s − τ). Combining these terms with the EX_above integral then gives:

${\text{EX}}_{\text{below}}+{\text{EX}}_{\text{above}}={\displaystyle {\int }_0^{\tau }\left[B\left(\tau +x\right)-2B\left(\tau \right)+B\left(\tau -x\right)\right]e^{-x}\hspace{0.17em}dx}\left(\text{SX},\tau <{\tau }_s/2\right)$(3a)

$+{\displaystyle {\int }_{\tau }^{{\tau }_s-\tau }\left[B\left(\tau +x\right)-B\left(\tau \right)\right]e^{-x}\hspace{0.17em}dx}\left(\text{AX},\tau <{\tau }_s/2\right).$(3b)

The integral in Eq. (3a) represents exchange between level τ and layers both above and below with equal optical thickness, so we refer to it as the symmetric exchange (SX) term. The integral in Eq. (3b) represents the residual, uncompensated heating from layers further below, which we refer to as the asymmetric exchange (AX) term. The regions contributing to these terms are shown schematically in Fig. 1a.

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Cartoon depicting the atmospheric layers relevant for the different cooling terms in Eq. (4), relative to a given layer at optical depth τ, for (a) τ < τ_s/2 and (b) τ > τ_s/2.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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Note that the SX integrand (3a) looks like a finite-difference approximation for a second derivative; in fact, if B is linear in τ then SX vanishes, because the cooling from the layer of depth τ above exactly cancels the heating from the layer of depth τ below. This difference of differences is the “double cancellation” described above, and in fact occurs in gray models of pure radiative equilibrium (section 3). Furthermore, in the limit that the SX term can indeed be approximated by a second derivative, we obtain the “diffusive” approximation to radiative cooling well known from textbooks (e.g., Pierrehumbert 2010; Goody and Yung 1989; Andrews et al. 1987). Note also that it is the SX term which yields strong radiative heating and cooling at the tropopause and stratopause respectively (e.g., Clough and Iacono 1995), as B has local extrema there and thus the EX_below and EX_above contributions do not compensate.

Returning to the EX_below and EX_above terms, if instead τ > τ_s/2 we then split the EX_above (rather than the EX_below) integral into an integral over the x intervals (0, τ_s − τ) and (τ_s − τ, τ), yielding

${\text{EX}}_{\text{below}}+{\text{EX}}_{\text{above}}={{\displaystyle \int }}_0^{{\tau }_s-\tau }\left[B\left(\tau +x\right)-2B\left(\tau \right)+B\left(\tau -x\right)\right]e^{-x}\hspace{0.17em}dx\left(\text{SX},\tau >{\tau }_s/2\right)$(3c)

$-{{\displaystyle \int }}_{{\tau }_s-\tau }^{\tau }\left[B\left(\tau \right)-B\left(\tau -x\right)\right]e^{-x}\hspace{0.17em}dx\left(\text{AX},\tau >{\tau }_s/2\right).$(3d)

The regions now contributing to SX and AX are shown in Fig. 1b. Note that AX now represents a cooling from layers above, rather than a heating from layers below. We will see below that this cooling leads to a failure of the CTS approximation near the surface.

These definitions of SX and AX then yield a new decomposition of radiative flux divergence, given by

${\displaystyle \frac{dF}{d\tau }}=\text{CTS}+\text{SX}+\text{AX}+\text{GX}.$(4)

This decomposition will be used throughout the paper.

3. Pure radiative equilibrium and the CTS approximation

We now apply the decompositions (1) and (4) as well as the validity criterion (6) to a gray gas in pure radiative equilibrium. We do this to provide a simplified context in which to compare decompositions, and also to better understand how the CTS approximation breaks down, resolving the paradox highlighted in question 2 from the introduction.

We now apply the old decomposition (1) to the solution (7) for B(τ), integrating where necessary to obtain analytical expressions for the various components of the flux divergence:

$\begin{array}{l}\text{CTS}=-{\displaystyle \frac{\text{OLR}}{2}}\left(1+\tau \right)e^{-\tau },\\ {\text{EX}}_{\text{above}}=-{\displaystyle \frac{\text{OLR}}{2}}\left[1-\left(1+\tau \right)e^{-\tau }\right],\\ {\text{EX}}_{\text{below}}={\displaystyle \frac{\text{OLR}}{2}}\left[1-\left({\tau }_s-\tau +1\right)e^{-\left({\tau }_s-\tau \right)}\right],\\ \text{GX}={\displaystyle \frac{\text{OLR}}{2}}\left({\tau }_s-\tau +1\right)e^{-\left({\tau }_s-\tau \right)}.\end{array}$(8)

These terms add to 0, as they must, and are plotted for τ_s = 20 in Fig. 2a. The CTS and GX terms behave equally and oppositely at their respective boundaries, with the discontinuity B_s − B(τ_s) at the surface yielding a source term jump equivalent to the jump between the atmosphere and space. The EX_above and EX_below terms cancel throughout most of the atmosphere, but decline toward the boundaries as the optical thickness of the relevant exchange layers declines to 0. In this picture, the CTS term does not dominate even for τ ~ O(1) due to cancellation by EX_below, though EX_below itself is partially canceled by EX_above.

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(a) Old and (b) new decompositions, Eqs. (8) and (9), for the gray gas PRE solution (7) for τ_s = 20. The CTS approximation (2) fails for τ ~ O(1), due to the presence of the uncompensated asymmetric exchange term AX [see (b)]. Note that the cancellation of EX_above and EX_below in (a) is captured implicitly by SX = 0 in (b).

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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Thus, we may answer question 2 from the introduction as follows: In PRE the double cancellation argument holds perfectly (i.e., SX = 0), but only to the extent that there is appreciable optical depth both above and below a given layer (which suppresses AX). This assumption is implicit in the double cancellation heuristic and can indeed hold throughout much of the atmosphere, but will fail near the boundaries, where there is indeed significant AX heating. In terms of our criteria (6), the criterion d²B/dτ² ≪ B holds but dB/dτ ≪ B does not, as Eq. (7) shows that dB/dτ ~ B, at least for τ ~ O(1) where the CTS term is significant.

4. Radiative–convective equilibrium and the CTS approximation

5. Gray RCE, pressure coordinates, and the τ = 1 law

To further test Eq. (17), as well as gain insight into the transition from PRE to a CTS-dominated RCE, Figs. 3a–c shows the decomposition (4) as applied to gray PRE as well as the gray RCE profile (13) with γ = 0.5 and 0.1. (All cases have α = 4 and τ_s = 20. In PRE we set β = 2, and for RCE we set T_s = 300 K, Γ = 7 K km⁻¹, and β is determined by γ. The OLR in PRE is taken to be the average of the OLR from the two RCE cases.) For γ = 0.5 we find that, unlike the PRE case, AX near τ = 1 is somewhat suppressed relative to CTS, due to the factor of γ in Eq. (15a), and that AX and GX are both strongly suppressed near the surface, due to both γ as well as the factor of 1/τ in Eq. (15a) (which at the surface is 1/τ_s = 1/20). This 1/τ factor is due physically to the decrease of dB/dτ with τ (because γ < 1), which suppresses the exchange terms, particularly near the surface when τ_s ≫ 1.

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(a)–(c) Decomposed and normalized ∂_τF profiles for PRE, RCE with γ = 0.5, and RCE with γ = 0.1. In RCE the exchange terms are suppressed by factors of γ, and further suppressed near the surface by factors of 1/τ [Eq. (15)]. The CTS term dominates as γ → 0, consistent with Eq. (17). (d)–(f) As in (a)–(c), but in pressure coordinates. Surface terms are now enhanced relative to (a)–(c), and maxima emerge near τ = 1 (τ = 1 marked by dashed gray lines). See text for discussion.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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The above effects are all further enhanced for the γ = 0.1 case. Here, the exchange terms essentially disappear near the surface, and are very small at lower τ, except for the AX term for τ ≪ 1. In this case the CTS approximation is very good indeed, in agreement with Eq. (17).

Though Figs. 3a–c and the discussion above paint a picture for how the CTS term comes to dominate in RCE, the profiles in Figs. 3a–c do not have a familiar shape, and in particular do not obey the usual τ = 1 law, according to which cooling profiles exhibit maxima at τ = 1 (e.g., Huang and Shahabadi 2014; Wallace and Hobbs 2006; Petty 2006; see also appendix B). Indeed, the maxima of the CTS term (1b) for PRE [Eq. (9)] and RCE [Eq. (13)] occur at τ = 0 and τ = γ, respectively.

In addition to the emergence of a τ = 1 maximum, another consequence of the pressure coordinates used in Figs. 3d–f is the relative enhancement of the exchange terms near the surface, as compared to the ∂_τF profiles in Figs. 3a–c. This is again due to the factor of dτ/dp ~ τ, which is enhanced near the surface when τ_s ≫ 1. This surface enhancement of radiative cooling in pressure coordinates reveals a limitation of both the CTS approximation, as well as our criteria in Eqs. (6) and (17): even when the CTS term dominates near τ = 1, near the surface the CTS term will be suppressed by $e^{-{\tau }_s}$ and will typically be much smaller than AX and GX there (this effect is not captured by the scalings (5) because these scalings ignored this exponential transmissivity factor). In RCE with γ ≪ 1 and in τ coordinates this failure of the CTS approximation is hidden as all terms are small near the surface, compared to the value of the CTS term at τ = 1 (Fig. 3c). In p coordinates, however, the surface enhancement of radiative cooling brings this failure of the CTS approximation to light (Fig. 3f). This breakdown of the CTS approximation for heating rates near the surface was noted earlier by Joseph et al. (1976), and will also be evident in the more realistic heating rates we compute in the next section.

6. Application to real greenhouse gases

The results so far have only been for idealized gray gases obeying the simple relationships (11) and (12). In this section we perform line-by-line radiative transfer calculations for H₂O and CO₂ to test the validity of our results for realistic greenhouse gases. In this section we will use pressure coordinates and apply our decomposition (4) to spectrally resolved heating rates ${\mathcal{H}}_{\tilde{\nu }}$, which depend on wavenumber $\tilde{\nu }$ (units of cm⁻¹) and will be plotted in units of K day⁻¹ cm⁻¹.

b. RFM results

Before examining the RFM results, let us evaluate the criterion (17) for H₂O and CO₂ in our idealized RCE profiles. To evaluate γ we need to estimate α and β from Eqs. (12) and (11). For α, note that $\alpha =\partial \hspace{0.17em}\ln B\left(\tilde{\nu },T\right)/\partial \hspace{0.17em}\ln T$ varies with both $\tilde{\nu }$ and T (Fig. 4), but is about 3 near $\left(\tilde{\nu },T\right)=\left(550\hspace{0.17em}{\text{cm}}^{-1},\hspace{0.17em}260\hspace{0.17em}\text{K}\right)$, a typical value for the H₂O rotation band (Jeevanjee and Fueglistaler 2020). For $\left(\tilde{\nu },T\right)=\left(650\hspace{0.17em}{\text{cm}}^{-1},\hspace{0.17em}260\hspace{0.17em}\text{K}\right)$ near CO₂ band center, α is closer to 4. We thus set ${\alpha }_{{\text{H}}_2\text{O}}=3$ and ${\alpha }_{{\text{CO}}_2}=4$. As for β, a noncondensable, pressure-broadened, well-mixed greenhouse gas such as CO₂ in Earth’s atmosphere is well known to have ${\beta }_{{\text{CO}}_2}=2$ (Pierrehumbert 2010). For H₂O, Jeevanjee and Fueglistaler (2020) estimated β and found that it varies somewhat in the vertical but has a typical value⁵ of ${\beta }_{{\text{H}}_2\text{O}}=5.5$. Plugging all of this as well as Γ = 7 K km⁻¹ into Eq. (14c) yields

$\begin{array}{l}{\gamma }_{{\text{H}}_2\text{O}}=0.1,\\ {\gamma }_{{\text{CO}}_2}=\mathrm{0.4.}\end{array}$(20)

According to Eq. (17) and Fig. 3, this suggests that the CTS approximation will hold quite well for H₂O, and less so for CO₂ (at least near τ = 1). Indeed, decomposed cooling profiles from our RFM calculations for wavenumbers with τ_s ≈ 20 confirm this (Fig. 5), and furthermore resemble the profiles from our gray RCE calculation (Figs. 3e,f).

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Contour plot of $\alpha \equiv \partial \hspace{0.17em}\ln B\left(\tilde{\nu },T\right)/\partial \hspace{0.17em}\ln T$, which appears as an exponent in Eq. (12).

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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Spectrally resolved heating rates ${\mathcal{H}}_{\tilde{\nu }}$ for CO₂ and H₂O, for wavenumbers with τ_s ≈ 20 ($\tilde{\nu }=710\hspace{0.17em}{\text{cm}}^{-1}$ and $\tilde{\nu }=428\hspace{0.17em}{\text{cm}}^{-1}$, respectively), decomposed according to Eq. (4). Gray dashed lines show τ = 1 level. These panels bear a resemblance to Figs. 3e and 3f, respectively, and suggest that the CTS approximation works well for H₂O but less so for CO₂. The sum of the terms in Eq. (4) computed offline (thin solid black line) compares very well to the heating calculated by differencing the net flux from RFM (black dashed line).

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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Of course, Fig. 5 only shows the decomposition (4) for one wavenumber for each gas. To test the robustness of our conclusions, Figs. 6 and 7 show each of the terms in Eq. (4) across the spectrum, as well as their spectral integrals, for H₂O and CO₂, respectively. These plots confirm that for H₂O the exchange terms are small and the CTS approximation is quite accurate, except near the surface. For CO₂, on the other hand, the exchange terms (and the AX term in particular) yield more substantial errors, at the surface and elsewhere. The SX term also provides the well-known tropopause heating from CO₂ (e.g., Thuburn and Craig 2002; Zhu et al. 1992). Physically, the CTS approximation holds well for H₂O because Clausius–Clapeyron scaling of H₂O concentrations yields a fairly rapid increase of τ with p (i.e., large β), and hence a small source function gradient dB/dτ, suppressing the exchange terms.

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(a)–(e) The decomposition of ${\mathcal{H}}_{\tilde{\nu }}$, as in Fig. 5, but now for all wavenumbers $\tilde{\nu }$ and for H₂O only. (f) Corresponding spectral integrals. This figure confirms that for H₂O the CTS approximation is quite accurate and the exchange terms are small, except near the surface. In (a)–(e) averages over spectral bins of width 10 cm⁻¹ are shown for clarity (results are not sensitive to this binning). Also shown in (a)–(e) are the τ = 1 contours (gray dashed line), and (c) and (f) show the tropopause height (horizontal gray dotted line).

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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As in Fig. 6, but for CO₂. The color bar is identical up to an overall scaling. In this case the exchange terms are more significant and the CTS approximation less accurate, as predicted by Eqs. (17) and (20). Note the strong contribution of the SX term at the tropopause (gray dotted line).

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0352.1

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7. Summary and discussion

We summarize our results as follows:

• We present a new decomposition of radiative flux divergence [Eq. (4)] which better captures the cancellation of exchange terms (Fig. 2).

• We derive the criterion γ ≪ 1 for the validity of the CTS approximation, near τ = 1 and in the case τ_s ≫ 1. Exchange terms are suppressed by factors of γ and 1/τ [Eq. (15)], though these terms can manifest near the surface in p coordinates (Fig. 3).

• We find that ${\gamma }_{{\text{H}}_2\text{O}}=0.1$ and ${\gamma }_{{\text{CO}}_2}=0.4$, consistent with the CTS approximation being quite accurate for H₂O but less so for CO₂, at least away from the surface (Figs. 5–7).

Let us begin with our simplified temperature profile. Generalizing our results to a nonisothermal stratosphere should be straightforward, as stratospheres with roughly constant lapse rates can also be modeled using Eq. (10). Negative lapse rates typical of the stratosphere will yield γ < 0 and thus could change the sign of the various exchange terms [cf. Eq. (15)], but we expect that generalizing Eq. (17) to |γ| ≪ 1 would still yield a criterion for the CTS approximation to hold. Furthermore, the small lapse rates typical of the stratosphere should favor the validity of the CTS approximation. For example, for CO₂ in a stratosphere with Γ = −2 K km⁻¹, Eq. (14c) yields γ = −0.12. This seems consistent with the validity of the CTS approximation for stratospheric CO₂ (e.g., Rodgers and Walshaw 1966, their Figs. 2a,c).

At the same time, more realistic temperature profiles can also have strong vertical gradients in Γ, arising, for example, at boundary layer or trade inversions, or at the tropopause or stratopause. This will likely lead to strong SX heating/cooling (because this term measures curvature in the temperature profile), and hence to a localized breakdown in the CTS approximation. Such a strong SX contribution can be seen at the tropopause for CO₂ in Fig. 7, and also occurs at the stratopause (e.g., Clough and Iacono 1995; Zhu et al. 1992).

Besides simplified temperature profiles, we also assumed constant RH. Relaxing this would add additional vertical variation to β = d lnτ/d lnp for H₂O, but should not affect γ or the CTS approximation in any other way. Other impacts of variable RH, unrelated to the CTS approximation, are discussed in Jeevanjee and Fueglistaler (2020). Convenient cases for studying these effects from more realistic temperature and humidity profiles might be those used in the Continual Intercomparison of Radiation Codes (CIRC; Oreopoulos and Mlawer 2010).

As for continuum absorption from H₂O, this should increase β since τ(p) for such wavenumbers will depend quadratically on vapor density (because continuum pressure-broadening is largely self-broadening, e.g., Pierrehumbert 2010), and such τ(p) will thus have an enhanced Clausius–Clapeyron dependence and thus be even more bottom heavy. This larger β would lead to a smaller γ, although it is unclear whether the CTS term would totally dominate at such wavenumbers. This is because such wavenumbers also tend to have τ_s ~ O(1) (at least in the present-day tropics), and thus the CTS term will overlap with the AX surface cooling already discussed. Also note that absolute values of lower-tropospheric H₂O cooling rates are strongly influenced by the continuum (Jeevanjee and Fueglistaler 2020), a caveat which should be kept in mind when interpreting Fig. 6.

Another influence on the strength of tropospheric cooling rates is H₂O–CO₂ overlap, which we have also neglected. How, then, should our conclusion that the CTS approximation applies well to H₂O, but not as well to CO₂, be applied to Earth’s atmosphere which contains both? Evaluation of the CTS approximation, as in Fig. 6f but in the presence of both H₂O and CO₂, shows that while the presence of CO₂ noticeably reduces tropospheric cooling rates, there is no noticeable degradation of the CTS approximation itself relative to the H₂O-only case (not shown). This is because the reduction in tropospheric cooling due to CO₂ is due to a replacement of tropospheric H₂O cooling by largely stratospheric CO₂ cooling (which is of course also how CO₂ lowers OLR). The actual contribution of CO₂ to tropospheric cooling is quite small relative to H₂O (cf. Figs. 6f and 7f), so errors in the CTS approximation to this small contribution are negligible.

In addition to studying more realistic cases, future work could also analyze the validity of the CTS approximation for other terrestrial GHGs, such as ozone or methane. Such cases may have different α and β values, and hence different γ. The 1300-cm⁻¹ band of methane, for instance, will have α much larger than 3 or 4 (Fig. 4). Furthermore, the parameterization (11) of a gas’s vertical distribution may not be appropriate for some gases, as in the case of ozone which is neither well mixed nor bottom heavy.

Going further afield, one might also apply the decomposition (4) to understand and model radiative cooling on other worlds, both within our solar system as well as exoplanets (e.g., Amundsen et al. 2014; Dufresne et al. 2005). If the CTS approximation holds, this might lead to simplified understanding as well as reductions in computational expense, similar to those leveraged in the past for terrestrial radiation (e.g., Schwarzkopf and Fels 1991; Fels and Schwarzkopf 1975).