a. Basic equations and the limiting case of the single layer model

The equivalent blackbody emission ${\mathrm{ℬ}}_e$ of the atmosphere is the mean value of the blackbody emission $\mathrm{ℬ}\left(P\right)$ weighted by the ω(P) function [Eq. (8b)]. For optically very thin atmospheres (τ_s ≪ 1), ${\mathrm{ℬ}}_e$ is equal to the pressure-weighted mean of $\mathrm{ℬ}\left(P\right)$. Then ${\mathrm{ℬ}}_e\approx \left[\mathrm{ℬ}\left(P_t\right)+\mathrm{ℬ}\left(P_s\right)\right]/2$ and is the same for the two idealized atmospheric profiles considered in this section (appendix A).

When the troposphere is optically thin, the radiance $\mathrm{ℱ}$ at the tropopause decreases when the optical thickness τ_s of the troposphere increases, starting from a value $\mathrm{ℱ}={\mathrm{ℬ}}_s$ when τ_s = 0 (Fig. 3, black line). As long as τ_s ≪ 1, the decrease of the radiance $\mathrm{ℱ}$ at the tropopause is proportional to τ_s and similar for both atmospheric profiles because the pressure-weighted mean temperature of both tropospheres is the same (see section a in appendix A). When the troposphere is isothermal, this decrease gradually slows down from optical thickness τ_s larger than 0.5 and reaches a plateau when the optical thickness is larger than about 4. When the troposphere is nonisothermal, the slowdown is not as fast as for the isothermal case and the decrease continues for optical thickness larger than 4. The limiting value of the radiance at the tropopause for infinite value of the optical thickness is much smaller (and therefore the greenhouse effect much higher) in the nonisothermal case compared to the isothermal case.

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Radiance at the tropopause [black: total, $\mathrm{ℱ}$; blue: emitted by the surface, ${\mathrm{ℱ}}_s$; red: emitted by the troposphere, ${\mathrm{ℱ}}_a$; Eq. (6)], weighted average blackbody emission of the troposphere [magenta: ${\mathrm{ℬ}}_e$; Eq. (8b)], as a function of the total optical thickness τ_s of the troposphere for an isothermal vertical profile (dashed line) and a profile for which the temperature decreases with altitude (solid line). The idealized troposphere is 12.6 km high with uniform absorption coefficient and volumetric mass density that only depends on pressure (see section 2a).

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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The model generally used in simplified explanations of the greenhouse effect [Eq. (1)] assumes that the troposphere is isothermal along the vertical. With this assumption, the flux at the tropopause does not decrease any more when the total optical thickness τ_s increases if τ_s is larger than 4. It is then said that the greenhouse effect “saturates.” This saturation effect almost disappears when the temperature decreases with height: the greenhouse effect continues to increase when the optical thickness τ_s increases, even for large values of τ_s. For a nonisothermal troposphere the altitude where the emitted radiation escapes to space matters. We now present how this effect of emission height can be quantified.

b. Contribution of absorptivity and emission height to radiance changes

The sensitivity of the radiance $\mathrm{ℱ}$ at the tropopause to a fractional change in amount of absorbing gases reads, according to Eq. (8):

${\displaystyle \frac{\partial \mathrm{ℱ}}{\partial f}}\equiv {\mathrm{ℱ}}^{\prime }={\displaystyle \frac{\partial {\mathit{Ƭ}}_s}{\partial f}}\left({\mathrm{ℬ}}_s-{\mathrm{ℬ}}_e\right)+\left(1-{\mathit{Ƭ}}_s\right){\displaystyle \frac{\partial {\mathrm{ℬ}}_e}{\partial f}},$(9)

which we rewrite as

${\mathrm{ℱ}}^{\prime }={\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }+{\mathrm{ℱ}}_{Z_e}^{\prime },$(10a)

${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }={\displaystyle \frac{\partial {\mathit{Ƭ}}_s}{\partial f}}\left({\mathrm{ℬ}}_s-{\mathrm{ℬ}}_e\right),\text{and}$(10b)

${\mathrm{ℱ}}_{Z_e}^{\prime }=\left(1-{\mathit{Ƭ}}_s\right){\displaystyle \frac{\partial {\mathrm{ℬ}}_e}{\partial f}}.$(10c)

These three terms are shown in Fig. 4 as a function of the total optical thickness τ_s of the troposphere.

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Sensitivity ${\mathrm{ℱ}}^{\prime }$ of the radiance at the tropopause to a fractional change in absorbing gas concentration as a function of the total optical thickness τ_s of the troposphere. Results are shown for the same isothermal (dashed line) and idealized decreasing temperature (solid line) profiles as in Fig. 3. The sensitivity ${\mathrm{ℱ}}^{\prime }$ (black) is decomposed in a contribution due to change in absorptivity (${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$; blue line) and a contribution due to change in emission height (${\mathrm{ℱ}}_{Z_e}^{\prime }$; red line). For the isothermal profile, ${\mathrm{ℱ}}^{\prime }$ is not visible because it is covered by ${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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According to Eq. (10b), ${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$ quantifies how much the radiance $\mathrm{ℱ}$ at the tropopause is directly impacted by a change in the transmissivity ${\mathit{Ƭ}}_s$, and therefore by a change in the absorptivity ${\mathcal{A}}_s=1-{\mathit{Ƭ}}_s$, when the amount of absorbing gases changes. The term ${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$ is the sensitivity of the radiance at the tropopause if the troposphere is isothermal, or would be isothermal, at a temperature that corresponds to a blackbody emission ${\mathrm{ℬ}}_e$ (i.e., $\partial {\mathrm{ℬ}}_e/\partial f=0$). For both temperature profiles, the absolute value of ${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$ linearly increases with τ_s, is maximum for τ_s ≈ 1, becomes very small for τ_s larger than 4, and is almost zero when the troposphere is fully opaque (Fig. 4). Note that ${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$ is slightly higher for the nonisothermal profile because ${\mathrm{ℬ}}_e$ has a smallest value with this profile as compared with the isothermal profile (Fig. 3).

The term ${\mathrm{ℱ}}_{Z_e}^{\prime }$ quantifies how much the radiance at the tropopause is impacted by a change in ${\mathrm{ℬ}}_e$ when the amount of absorbing gases changes. Radiance ${\mathrm{ℬ}}_e$ [Eq. (8b)] is the weighted average of the Planck function over the whole troposphere with a weight ω(P) [Eq. (8c)], which depends on the optical exchange factor between the atmosphere at pressure P and the tropopause (Dufresne et al. 2005). This weight varies from a function that is constant with pressure when the total optical thickness is low (τ_s ≪ 1) to a function that is maximum at the tropopause, decreases with increasing pressure and is almost zero close to the surface when the total optical thickness is large (τ_s ≫; 1) (Fig. 5; section b in appendix A). As a consequence, the radiation that reaches the tropopause is emitted on average at lower pressure (i.e., at higher altitude) when the optical thickness of the troposphere increases. It is said that the “emission height” increases (Hansen et al. 1981; Held and Soden 2000; Pierrehumbert 2010; Archer 2011; Benestad 2017). The variable ${\mathrm{ℱ}}_{Z_e}^{\prime }$ quantifies how much this change in emission height impacts the radiance at the tropopause. It is zero for an isothermal troposphere since $\left(\partial {\mathrm{ℬ}}_e/\partial f\right)=0$. If the temperature of the troposphere decreases with height, an increase of emission height yields a decrease of the temperature, a decrease of the Planck function and therefore a decrease of the upward radiance at the tropopause. The sensitivity ${\mathrm{ℱ}}_{Z_e}^{\prime }$ due to change in emission height increases with the total optical thickness τ_s of the troposphere, reaches a maximum for τ_s ≈ 4, and then slowly decreases (red line in Fig. 4).

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Vertical profile of ω(P) for different values of the total optical thickness τ_s of the troposphere (black: 0.01, blue: 1, red: 2, green: 4, and magenta: 8) for the same idealized troposphere with a uniform absorption coefficient as in Fig. 3; ω(P) is the normalized optical exchange factor between the troposphere at pressure P and the tropopause [Eq. (8c)].

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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c. Emission height

After defining the contribution of the change in emission height to the change in radiance at the tropopause, we now define the emission height itself. Since we assume in this section that the absorption coefficient k is constant, the optical thickness increases linearly with pressure [Eq. (5)]. Therefore, many radiative variables are easier to compute and to interpret in pressure coordinate rather than in altitude coordinate. We will therefore continue to write the equations in pressure coordinate, and the “emission height” will be defined as the altitude corresponding to the “emission pressure.”

When the troposphere is isothermal, $\mathrm{ℬ}\left(P\right)={\mathrm{ℬ}}_e$ and therefore Ω(P) = ω(P). The probability density that a photon emitted by the troposphere and that reached the tropopause has been emitted at a level of pressure P is equal to the probability density that a photon going downward at the tropopause is absorbed at level of pressure P. This is consistent with the reciprocity principle. Therefore, when the total optical thickness τ_s is small, the emission pressure is equal to the average between the tropopause pressure and the pressure at surface, that is, (P_t + P_s)/2. The corresponding altitude is slightly higher than 4 km, which is consistent with what is observed in Fig. 6. Relative to the isothermal profile, Z_e is lower when the temperature decreases with height because the Planck function gives more weight to the lower and warmer part of the troposphere.

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Emission height Z_e (black lines) as a function of the total optical thickness τ_s for a troposphere with an idealized decreasing temperature profile as in Fig. 3 (solid line) and change ΔZ_e of this emission height when the amount of absorbing gas is doubled (dotted line). The dashed line displays the emission height Z_e for the isothermal profile. Shades display the function τ(τ_s, z), defined as the optical thickness τ at altitude z when the total optical thickness of the troposphere is τ_s. For instance, the emission height Z_e almost coincides with the isoline τ(τ_s, z) = 1 for optically thick atmospheres (τ_s > 4).

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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Starting from the altitude where the pressure is (P_t + P_s)/2, the emission height Z_e increases when the total optical thickness τ_s increases (Fig. 6). The emission height Z_e is commonly approximated as the altitude where the optical thickness is one (Pierrehumbert 2010; Huang and Bani Shahabadi 2014). For the profiles considered here, this approximation is valid as soon as the total optical thickness is larger than about 4 (Fig. 6).

a. Single absorption line

The sensitivity of the spectral radiance at the tropopause to a fractional change in the absorbing gas for a line can be deduced from single frequency results (Fig. 4). A first example is shown for a single and weak absorption line (Fig. 7). The optical thickness at the absorption line center is about 0.75 and decreases rapidly away from the line center. The sensitivity of the radiance is maximum at the line center and is primarily due to the change in absorptivity (${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$; blue line) as the optical thickness is small.

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(top) Total optical thickness τ_s and (bottom) sensitivity of the radiance at the tropopause to a fractional change in absorbing gases as a function of wavenumber ν for a single weak absorption line and the MLS atmospheric profile. This sensitivity (${\mathrm{ℱ}}^{\prime }$; black line) is decomposed into the contribution due to the change in emission height (${\mathrm{ℱ}}_{Z_e}^{\prime }$; red line) and due to the change in absorptivity (${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$; blue line). The abscissa is the distance from the line center (ν_c = 550 cm⁻¹); the line width is 0.1 cm⁻¹ near the surface.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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This picture is very different for a line whose absorption intensity is 12 times larger and that will be referred later as a line of “medium intensity” (Fig. 8). Around the absorption line center, the sensitivity ${\mathrm{ℱ}}_{\mathit{Ƭ}}^{\prime }$ due to a change in absorptivity is zero as one may expect from Fig. 4. In this spectral region the sensitivity ${\mathrm{ℱ}}_{Z_e}^{\prime }$ due to a change in emission height is the dominant factor. Close to the absorption line center, the optical thickness is large and the mean emission height is located close to the tropopause (Fig. 9). An increase of optical thickness has little impact on the emission height. Away from the absorption line center, the sensitivity due to a change in emission height is still the dominant factor and an increase of absorbing gas decreases the radiance at the tropopause and therefore increases the greenhouse effect. Further away from the absorption line center, the sensitivity due to a change in total absorption dominates, and slowly decreases away from the absorption line center.

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As in Fig. 7, except that the intensity of the line is larger by a factor of 12, referred as the line of medium intensity.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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Emission height Z_e (black solid line) as a function of the distance from the absorption line center for the same conditions as in Fig. 8, and change ΔZ_e of this emission height when the amount of absorbing gas is doubled (dotted line). Color shading indicates the optical thickness τ of the troposphere at altitude z and wavenumber ν.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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b. Change in emission height when doubling the amount of absorbing gas

As a benchmark the amount of absorbing gas is doubled over the whole atmospheric profile. For the idealized atmosphere, the change ΔZ_e in emission height is zero when the optical thickness τ_s of the tropopause is zero, as one may expect (Fig. 6, dotted line). It increases with τ_s up to more than 2 km for τ_s ≈ 4 and then decreases with increasing τ_s because the emission height is already close to the tropopause.

For the weak absorption line, the change in emission height is small. It varies from about 20 m at 0.3 cm⁻¹ from the absorption line center to about 800 m at the absorption line center (not shown). For the absorption line of medium intensity (Fig. 9), the change in emission height increases from about 200 m at 0.3 cm⁻¹ from the absorption line center up to 2 km at a wavenumber for which the optical thickness is about 4 and finally decreases to 750 m at the absorption line center.

From these results, one may expect that for a doubling of the CO₂ concentration, the change in emission height will have a value varying from a few tens of meters in spectral regions where the absorptivity is either very weak or very strong to a maximum value of ≈1–2 km in spectral regions where the optical thickness is a few units (τ_s ≈ 2–8).

The results presented until now can be easily reproduced and provide the basis to understand the key phenomena that drives the greenhouse effect. In the next section we will use this understanding to interpret results produced by a comprehensive reference radiation code.

a. Framework for radiative flux

b. A reference line-by-line model based on a net exchange formulation

The line-by-line radiative model we use is presented in Eymet et al. (2016) and its main originality is to rely on the NEF. In a first step (Kspectrum code), a synthetic high-resolution (typically 0.0005 cm⁻¹) absorption spectrum is computed for the required atmospheric profile using the HITRAN 2012 molecular spectroscopic database (Rothman et al. 2013) with Voigt line profiles. For CO₂, sub-Lorentzian corrections are taken into account. For H₂O, the CKD continuum is used with a 25-cm⁻¹ truncation and removing the “base” of each transition (Clough et al. 1989; Mlawer et al. 2012). In a second step (HR_PPart code), radiative transfer is computed based on 1D (over a single line of sight) or 3D (angularly integrated) analytical expressions of spectral radiative net exchange rates and spectral radiative fluxes. We compute the radiative forcing for CO₂ and H₂O changes on the basis of the experiments defined in Collins et al. (2006): the reference experiment, which is the MLS atmospheric profile with a CO₂ concentration of 287 ppmv (experiment 1a), an experiment in which the CO₂ concentration is doubled (experiment 2b), and an experiment in which the CO₂ concentration is doubled and the concentration of H₂O is increased by 20% (experiment 4a). In this example, the only absorbing gases considered are H₂O, CO₂, and ozone, and the troposphere is discretized into 31 vertical layers. The results compare well to those published by Collins et al. (2006), as shown in Table 1.

Table 1.

Difference of the net flux (W m⁻²) at the TOA, at 200 hPa [Δ$\overline{F}$(200)], and at the surface (Δ$\overline{F}$_s) for a CO₂ doubling and for an increase of H₂O by 20% for the MLS atmospheric profile and for the thermal infrared spectral interval (100–2500 cm⁻¹; i.e., 4–100 μm). The results computed with our model (label KS) are compared with those published by Collins et al. (2006) (label C06) for an ensemble of line-by-line models using the same atmospheric profiles (see section 4b).

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c. Results for a realistic atmospheric profile

We use the same MLS atmospheric profile and consider only the troposphere, from the surface (P_s = 1013 hPa, H_s = 0 m, and T_s = 294 K) to the tropopause (P_t = 190 hPa, H_t ≈ 12.6 km, and T_t = 218.4 K), as presented above.

We first focus on two CO₂ weak absorbing lines. In Fig. 10, and only in this figure, we exclude absorption by the H₂O continuum in order to have an optical thickness that is as small as possible. For the weaker absorption line for which the optical thickness is always less than one (Fig. 10, left column), the shape of the optical thickness resembles that of the idealized one (Fig. 7). The optical thickness is low (τ_s ≪ 1) and the emission height is about 2–3 km, as expected from Fig. 9. When doubling the CO₂ concentration, the change in optical thickness is almost equal to the value for the reference atmosphere, the difference is due to some absorption by H₂O. The change in emission height is less than 100 m at wavenumbers far away from the absorption line center and increases to a few hundred meters at the absorption line center. The change in the tropopause irradiance is largely dominated by the contribution of the change in absorptivity.

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(top) Total optical thickness τ_s of the troposphere (black line) and its change Δτ_s (magenta line) for a CO₂ doubling, (middle) emission height Z_e (black line) and its change ΔZ_e (magenta line) for a CO₂ doubling, and (bottom) sensitivity F′ of the flux at the tropopause (black line) to a fractional change in CO₂ and contributions of change in absorptivity (blue line: $F_{\widehat{\mathit{Ƭ}}}^{\prime }$) and in emission height (red line: $F_{Z_e}^{\prime }$) as a function of wavenumber (cm⁻¹). Shown are (left) a weak absorption line (964.5–965.1 cm⁻¹) and (right) an intermediate absorption line with a companion weak absorbing line (758.5–759.1 cm⁻¹).

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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For a more absorbing line with a companion weak absorbing line (Fig. 10, right column), the emission height is about 2–3 km far from the absorption line center, where the optical thickness is below 1. At the absorption line center, the emission height reaches 8 km, which is closer to the tropopause. When doubling the CO₂ concentration, the change in emission height is a few hundred meters far from the absorption line center to more than a kilometer at the absorption line center. The change in the tropopause irradiance is dominated by the contribution of the change in emission height.

The results we obtained with the various idealized configurations are consistent with those we obtained with the reference model. The understanding we gained with the idealized examples can be applied to interpret the results with much more complex and realistic models.

We now consider the “thermal infrared” spectral interval from 100 to 2500 cm⁻¹ (4–100 μm). In Figs. 11 and 12 (and later in Fig. 14), variables are smoothed on a 10-cm⁻¹ spectral interval to make the figure more readable. The spectral dependencies of the radiative flux at the TOA, at the tropopause, within the atmosphere, and of the radiative cooling rate in the atmosphere, as well as how they change when changing the CO₂ concentration, have already been addressed in many studies (Kiehl and Ramanathan 1983; Kiehl 1983; Charlock 1984; Clough and Iacono 1995; Harries et al. 2001; Huang 2013). Mlynczak et al. (2016) show that these results were remarkably insensitive to known uncertainties in the main CO₂ spectroscopic parameters. Zhong and Haigh (2013) showed how the flux at the TOA varies over a wide range of CO₂ values, and they showed that the spectral response is very different depending on the CO₂ concentration. Here we consider the response for an atmosphere with a CO₂ concentration close to its preindustrial value (287 ppmv).

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(a) Optical thickness τ_s (black line: total optical thickness; magenta line: optical thickness due to CO₂), (b) emission height Z_e, (c) emission temperature T_e, and (d) upward radiative flux at the tropopause (black: total, $\mathrm{ℱ}$; blue: emitted by the surface, ${\mathrm{ℱ}}_s$; red: emitted by the troposphere, ${\mathrm{ℱ}}_a$) for the MLS atmospheric profile. The abscissa is given in wavenumber (cm⁻¹) at the bottom and in wavelength (μm) at the top. Variables are smoothed on a 10 cm⁻¹ spectral interval.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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Changes due to a CO₂ doubling (magenta line) and to an increase by 20% of the H₂O concentration (green line) of the (a) optical thickness τ_s, (b) emission height Z_e, and (c) emission temperature T_e for the same atmospheric profile as in Fig. 11. The changes ΔF of the flux at the tropopause (black line) and the contributions of the change in atmospheric absorptivity (blue line, $\text{Δ}{F}_{\widehat{\mathit{Ƭ}}}$) and in emission height (red line; $\text{Δ}{F}_{Z_e}$) are shown for (d) CO₂ and (e) H₂O. The abscissa is given in wavenumber (cm⁻¹) at the bottom and in wavelength (μm) at the top. Variables are smoothed on a 10-cm⁻¹ spectral interval.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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The total optical thickness τ_s (Fig. 11a, black line) is primarily due to H₂O absorption, except around 660 and 2300 cm⁻¹ where the two CO₂ strong absorption band systems at 15 and 4.3 μm (magenta) are dominant. The total optical thickness varies over many orders of magnitude, from about 1 in the atmospheric window (between 800 and 1200 cm⁻¹) to 10⁴–10⁵ in the H₂O and CO₂ absorption bands. When the data are not smoothed, the range is even larger, from a few tenths up to 10⁶.

The emission height (Fig. 11b) almost increases with the logarithm of the total optical thickness τ_s (Huang and Bani Shahabadi 2014). It varies for 2 km in the atmospheric window up to 12 km (i.e., almost the tropopause height) in the spectral region where the optical thickness is very high, especially for the CO₂ bands. For the same optical thickness, the emission height in the CO₂ absorption bands are larger than for the H₂O absorption bands as the CO₂ concentration is uniform over the whole troposphere whereas the H₂O concentration strongly decreases with height.

We define the emission temperature as the temperature for which the Planck function is equal to B_e defined by Eq. (20b). The emission temperature (Fig. 11c) directly follows the evolution of the emission height. The dependence is about 7 K km⁻¹, as one may expect from the value of the temperature gradient in the troposphere. The upward flux at the tropopause may have been emitted either from the surface or from the troposphere [Eqs. (17) and (6)]. One can see in Fig. 11d that almost all the flux at the tropopause has been emitted by the troposphere, except in the atmospheric window where the emissions by the surface and the troposphere contribute almost equally (Costa and Shine 2012).

The ozone absorption band around 1050 cm⁻¹ has a specific signature as ozone is mainly located in the higher part of the troposphere. This band has little impact on the optical thickness but has a visible signature on the emission height, the emission temperature, and the outgoing flux (Fig. 11).

Figure 12 displays the changes in total optical thickness, emission height, emission temperature, and upward flux at the tropopause when doubling the CO₂ concentration or when increasing the H₂O concentration by 20%. For the CO₂ 15-μm band system (660 cm⁻¹), the change in emission height, emission temperature, and tropopause flux is maximum on the edges of the band (Figs. 12b–d), where the CO₂ optical thickness is about a few units (Fig. 11a). In these spectral regions, the change in emission height is about 1 km and the change in emission temperature is about 7 K. The change in emission height is almost zero at the band center as the emission height is already close to the tropopause (i.e., close to the maximum height). The change in the flux at the tropopause is almost only due to the change in emission height (Fig. 12d). For the 4.3-μm (2300 cm⁻¹) CO₂ band, the changes in emission height and emission temperature resemble those for the 15-μm (660 cm⁻¹) band, but these changes have almost no impact on the tropopause flux as the Planck function is almost zero at these wavenumbers for the atmospheric temperature. In addition to these two very strong absorption bands, CO₂ also has some minor bands that produce small changes in emission height, emission temperature, and tropopause flux. In the spectral domain of these minor bands, the optical thickness is small (about 10⁻¹) and is due to absorption by both H₂O and CO₂. As a result, both the change in emission height and in absorptivity play a comparable role, whereas the change in absorptivity would have had a dominant role if CO₂ were the only absorbing gas. Note that this holds for the current atmosphere but not for an atmosphere with very high CO₂ concentration: these “minor” bands contribute to the CO₂ forcing by about 6% in current conditions, but they contribute by about 25% for CO₂ concentration that are 100 times larger (Augustsson and Ramanathan 1977; Zhong and Haigh 2013).

When the H₂O concentration is increased, the change in emission height is about 200 m (Fig. 12b) over spectral intervals that are much wider (100–600 and 1300–2000 cm⁻¹) than for CO₂. In these intervals the absorption by H₂O is strong and the change of the flux at the tropopause is almost only due to the change in emission height (Fig. 12e). In spectral regions where absorption by CO₂ dominates (600–750 cm⁻¹), the change in H₂O is completely masked by the CO₂ absorption. In most of the atmospheric window (750–1300 cm⁻¹), the change in emission height is small (<100 m) and the change of the flux at the tropopause is mainly due to the change in absorptivity, with a significant contribution of the water vapor continuum (Costa and Shine 2012). An exception is around 1050 cm⁻¹ where ozone absorbs. In this spectral region both the ozone and the water vapor emit radiation and the emission height includes both the contribution of ozone, which is mainly located in the high troposphere, and the contribution of water vapor, which is mainly located in the lower troposphere. When the H₂O concentration increases, the radiation emitted by H₂O that reaches the tropopause increases whereas the radiation emitted by ozone that reaches the tropopause does not change. As a result the emission height decreases by about 200 m (Fig. 12b), the emission temperature increases (Fig. 12c), and the contribution of the change in emission height to the flux at the tropopause is positive (Fig. 12e).

When considering the radiative flux over the whole thermal infrared domain, the decrease of the flux at the tropopause due to an increase of CO₂ is primarily due (by about 90%, Table 2) to the change in emission height, the change in absorptivity playing a minor role (about 10%). For an increase of water vapor, the change in absorptivity plays a more important role (about 40%) but the change in emission height still plays the dominant role (≈60%). However, this significant contribution of the change in absorptivity for H₂O is primarily due to the H₂O continuum. When the continuum is suppressed, the change in emission height is as high as 80% and the contribution of the change in absorptivity reduces to 20%.

Table 2.

Difference $\text{Δ}\overline{F}$ of the upward flux at the tropopause, difference $\text{Δ}{\overline{F}}_{Z_e}$ of this flux due to change in emission height, difference $\text{Δ}{\overline{F}}_{\widehat{\mathit{Ƭ}}}$ due to change in absorptivity, relative contribution of each of the changes ($\text{Δ}{\overline{F}}_{Z_e}$ and $\text{Δ}{\overline{F}}_{\widehat{\mathit{Ƭ}}}$) to the total $\text{Δ}\overline{F}$, and change $\text{Δ}{\overline{\widehat{\mathit{Ƭ}}}}_s$ of the broadband transmissivity of the atmosphere. The differences are computed for a CO₂ doubling (2b − 1a; first row) and an increase of H₂O by 20% (4a − 2b; second row) for the atmospheric profiles (Atm profile) presented in the text (section 4b).

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d. Including the stratosphere

So far and for simplicity we considered an atmosphere that extends from the surface to the tropopause, and therefore in which the vertical temperature gradient is always negative and driven by the convective adjustment. We now consider an atmosphere that extends to an altitude of 100 km and will show that the main results are still valid when the temperature adjustment of the stratosphere is taken into account.

In the stratosphere, the radiative cooling is compensated by the dynamic heating with a relaxation time of a few months. As the dynamics in the troposphere and in the stratosphere are weakly coupled, it has been shown (Hansen et al. 1981, 1997; Stuber et al. 2001; Forster et al. 2007) that it is more relevant to compute the radiative forcing after allowing stratospheric temperatures to adjust to a new radiative equilibrium than to compute the radiative forcing with a fixed stratospheric temperature. The stratospheric temperature adjustment is computed assuming no change in stratospheric dynamics as follows: after computing the radiative budget S₁(z) at each altitude z for the reference concentration and temperature, the radiative budget S₂(z) at each altitude z is computed with the same temperature profile but a modified CO₂ concentration. The temperature in the stratosphere is then adjusted until S₂(z) ≈ S₁(z) at each altitude z of the stratosphere. The results we obtain for the MLS profile and a doubling of the CO₂ concentration are shown in Fig. 13. By construction the temperature in the troposphere does not change. The temperature in the stratosphere decreases as expected (Hansen et al. 1997; Stuber et al. 2001), with a temperature cooling of 5–10 K.

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Temperature as a function of altitude for the full MLS atmospheric profile, i.e., including the stratosphere. Both the reference temperature (dashed line) and the temperature after the stratosphere has adjusted to a doubling of the CO₂ concentration (solid line) are shown.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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Relative to the troposphere-only case (Fig. 11), including the stratosphere increases the emission height in the center of the CO₂ absorption band systems at 15 and 4.3 μm where the emission height reaches values up to 30 to 40 km (Fig. 14a). This height is even larger when looking at the full resolution data (not shown). Including the stratosphere increases the optical thickness around the 9.7 μm (≈1050 cm⁻¹) O₃ absorption band by a few units, which has a significant impact on both the emission height and the flux at the TOA (Figs. 14a,b).

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For the MLS atmospheric profile including the stratosphere, (a) emission height Z_e and (b) radiative flux at the TOA in total (black), emitted by the surface (blue), and emitted by the atmosphere (red). Also shown is change in (c) emission height (ΔZ_e, magenta) and (d) the flux at the TOA if the temperature in the stratosphere is held fixed (dashed line) or is adjusted (solid line). The abscissa is given in wavenumber (cm⁻¹) at the bottom and in wavelength (μm) at the top. Variables are smoothed on a 10-cm⁻¹ spectral interval.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1

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When doubling the CO₂ concentration, the change in emission height (Fig. 14c) is comparable to the case without stratosphere (Fig. 12b) except in the 15- and 4.3-μm CO₂ absorption bands. At these band centers, the emission height can now be larger than the tropopause height; the increase in emission height is not blocked anymore and it has an almost constant value of about 3 km. One can show that the change in emission height is almost constant for a well-mixed absorption gas when absorption is saturated because the emission height is then close to the height where the optical thickness is equal to one (Fig. 6). This large change in emission height has a clear signature on the change of the flux at the TOA for the 15-μm CO₂ absorption band. At the absorption band center, a higher emission height leads to an increase of the outgoing flux because the temperature vertical gradient in the stratosphere is positive. However, this happens only if the temperature in the stratosphere is fixed (Fig. 14d, dashed line) as already shown (Kiehl 1983; Charlock 1984). If the temperature of the stratosphere is adjusted as explained above, the decrease of temperature in the stratosphere leads to a decrease of the emitted radiation. As a result, the change in the outgoing flux at the center of the 15-μm CO₂ absorption band is slightly negative. The pattern of the change of the spectral flux around the 15-μm CO₂ absorption band is similar if the atmosphere only extends up to the tropopause (Fig. 12d) and if the atmosphere extends higher than the tropopause but the stratospheric adjustment is considered (Fig. 14d, continuous line). At the first order, the interpretation of the results we obtained with an atmosphere reduced to the troposphere can be extended to a full atmosphere where the temperature of the stratosphere is adjusted. However, the adjustment of the stratosphere also impacts the emission by other gases: H₂O for wavenumbers lower than 500 cm⁻¹ and ozone near 1050 cm⁻¹.