1. Introduction
The broadband absorptivity of the atmosphere increases when the amount of water vapor increases, which supports the simple idea that an increase of atmospheric absorptivity in the infrared increases the greenhouse effect. However, the broadband absorptivity shows very little increase when the CO2 concentration increases, especially for regular amounts of water vapor (Fig. 1a). This is the well-known “saturation effect” of CO2 absorption (Archer 2011; Pierrehumbert 2011; Zhong and Haigh 2013), first pointed out by Ångström (1900), who questioned the results of Arrhenius (1896) showing the impact of CO2 concentration on the Earth surface temperature. It has been shown that the CO2 absorption is not fully saturated (Pierrehumbert 2011; Shine et al. 1995), and that a CO2 increase modifies both the broadband and the spectral flux at the TOA (Kiehl 1983; Charlock 1984; Harries et al. 2001; Mlynczak et al. 2016). This saturation argument is still used in the public debate to claim that an increase of CO2 concentration has very limited impact, if any, on the greenhouse effect.
(a) Broadband absorptivity
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
The “saturation paradox” can be summarized as follows: why does the greenhouse effect increase with the CO2 concentration (Fig. 1b) whereas the broadband absorptivity does not increase as much, especially when water vapor is present (Fig. 1a)? As highlighted by Eq. (2), the absorptivity is not the only main parameter that controls the greenhouse effect; the emission temperature Ta of the atmosphere is also fundamental. If the increase of CO2 concentration has little impact on absorptivity, it has a significant impact on Ta. When the CO2 increases, the infrared radiation that escapes toward space is emitted by the atmosphere at a higher altitude. As most of the radiation is emitted by the troposphere, higher altitude means lower emission temperature, a lower value of the Planck function, a lower value of the radiation emitted toward space, and therefore a higher value of the greenhouse effect (Hansen et al. 1981; Pierrehumbert 2010; Archer 2011; Benestad 2017). For a doubling of the CO2 concentration, the average value of the change in emission height is about 150 m, assuming that the radiative forcing of about ≈4 W m−2 can be translated into a change in blackbody temperature emission, and then into a change in emission height assuming a temperature vertical gradient of ≈6.5 K km−1 (Held and Soden 2000).
Beyond the single layer model, for fundamental physical reasons, the increase of the greenhouse effect due to an increase of the concentration of an absorbing gas, in particular CO2, is partly due to an increase of absorptivity and partly due to an increase of emission height (Pierrehumbert 2010). However, the contributions of each of these two effects have not been quantified yet, and the main goals of this paper are to present a framework that allows quantifying the contribution of these two effects and to perform the quantification. A second goal is to quantify the change in emission height, and not only its impact on the flux at the TOA. This offers the possibility to propose a new quantitative simplified description of the greenhouse effect that is more realistic than the too simple single layer model called the “blanket model” (Benestad 2017).
In this study, we will use only prescribed atmospheric profiles and will therefore compute the forcing when changing the absorbing gas concentration. All calculations are for cloudless skies. In section 2 we present the framework that allows us to separate and quantify the contribution of absorptivity and that of emission height to the flux at the tropopause, and therefore to the greenhouse effect. To allow some analytical developments, especially for simple limiting cases, we consider monochromatic radiances and idealized vertical atmospheric profiles. In section 3 we still consider radiances but with realistic atmospheric profiles. This will help us to interpret the results presented in section 4, where we compute the flux at the tropopause over the whole thermal infrared domain and where we independently increase the concentration of the two most important greenhouse gases on Earth: water vapor (H2O) and CO2. The temperature adjustment of the stratosphere is also analyzed. A summary and conclusions are given in section 5.
2. Formulation with simplified conditions
Vertical profile of the Planck function
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
a. Basic equations and the limiting case of the single layer model
The equivalent blackbody emission
When the troposphere is optically thin, the radiance
Radiance at the tropopause [black: total,
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
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 τs increases if τs 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 τs increases, even for large values of τs. 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
Sensitivity
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
According to Eq. (10b),
The term
Vertical profile of ω(P) for different values of the total optical thickness τs 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
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,
Emission height Ze (black lines) as a function of the total optical thickness τs for a troposphere with an idealized decreasing temperature profile as in Fig. 3 (solid line) and change ΔZe of this emission height when the amount of absorbing gas is doubled (dotted line). The dashed line displays the emission height Ze for the isothermal profile. Shades display the function τ(τs, z), defined as the optical thickness τ at altitude z when the total optical thickness of the troposphere is τs. For instance, the emission height Ze almost coincides with the isoline τ(τs, z) = 1 for optically thick atmospheres (τs > 4).
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
Starting from the altitude where the pressure is (Pt + Ps)/2, the emission height Ze increases when the total optical thickness τs increases (Fig. 6). The emission height Ze 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).
3. Results with more realistic cloudless atmospheres
We now abandon previous idealized vertical profiles and consider a realistic cloudless atmosphere, namely the midlatitude summer (MLS) atmospheric profile (McClatchey et al. 1972; Anderson et al. 1986) that has often been used to benchmark radiative codes (Ellingson et al. 1991; Collins et al. 2006; Pincus et al. 2015).
As in the previous section, we consider an atmosphere that ends at the tropopause. The temperature and the pressure at the surface and at the tropopause are close to the previous idealized profile (Ps = 1013 hPa, Hs = 0 m, Ts = 294 K, Pt = 190 hPa, Ht ≈ 12.6 km, and Tt = 218.4 K). The vertical profile of temperature is almost linear with altitude, and therefore the vertical profile of the Planck function at νc = 550 cm−1 is not linear with pressure anymore. The volumetric mass density varies according to the perfect gas law and the atmosphere is discretized into 65 vertical layers. The CO2 concentration is 287 ppmv as in Collins et al. (2006). We perform the same computations as in the previous section with this new profile, and the results show few differences relative to those displayed in Figs. 3–6 (not shown). The exact values are slightly modified but all the key features are identical.
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 (
(top) Total optical thickness τs 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 (
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
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
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
Emission height Ze (black solid line) as a function of the distance from the absorption line center for the same conditions as in Fig. 8, and change ΔZe 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
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 ΔZe in emission height is zero when the optical thickness τs of the tropopause is zero, as one may expect (Fig. 6, dotted line). It increases with τs up to more than 2 km for τs ≈ 4 and then decreases with increasing τs 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 (τs ≈ 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.
4. Radiative flux over the whole infrared domain with realistic radiative and thermodynamic properties
Until now we considered only radiances that allowed us to avoid angular integration. In this section we show how the framework based on radiances can be easily transposed to a framework for irradiance, or radiative flux. We use the classical approximations for pristine atmospheres. The atmosphere is absorbing and nonscattering, perfectly stratified along the horizontal (plane parallel assumption) and the surface has an emissivity of 1.
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.
Difference of the net flux (W m−2) at the TOA, at 200 hPa [Δ
c. Results for a realistic atmospheric profile
We use the same MLS atmospheric profile and consider only the troposphere, from the surface (Ps = 1013 hPa, Hs = 0 m, and Ts = 294 K) to the tropopause (Pt = 190 hPa, Ht ≈ 12.6 km, and Tt = 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 (τs ≪ 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.
(top) Total optical thickness τs of the troposphere (black line) and its change Δτs (magenta line) for a CO2 doubling, (middle) emission height Ze (black line) and its change ΔZe (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:
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
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).
(a) Optical thickness τs (black line: total optical thickness; magenta line: optical thickness due to CO2), (b) emission height Ze, (c) emission temperature Te, and (d) upward radiative flux at the tropopause (black: total,
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
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 τs, (b) emission height Ze, and (c) emission temperature Te 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,
Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0193.1
The total optical thickness τs (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 τs (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 Be 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%.
Difference
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.
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
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).
For the MLS atmospheric profile including the stratosphere, (a) emission height Ze 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 (ΔZe, 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
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.
5. Summary and conclusions
In this article we presented a framework that allows us to make a direct and precise link between the basic radiative transfer equations in the atmosphere on one hand, and the concept of emission height on the other hand. This allowed us to quantify how much a change in the greenhouse effect originates from a change in the emission height and how much originates from a change in the absorptivity of the atmosphere (i.e., the absorption over the entire height of the atmosphere).
The fact that a saturation of the absorptivity of the atmosphere leads to a saturation of the greenhouse effect is directly related to the hypothesis of an isothermal atmosphere. When this simplification is removed and the decrease of temperature with altitude is considered, as is the case in the troposphere, the greenhouse effect can continue to increase even if the absorptivity of the atmosphere is saturated.
The fundamental difference between our approach and other approaches such as the “bulk emission temperature” (Benestad 2017) or the “brightness temperature” commonly used in remote sensing, is that we split the radiation leaving the atmosphere toward space into two terms: the radiation that has been emitted by the surface [termed “surface transmitted irradiance” in Costa and Shine (2012)] and the radiation that has been emitted by the atmosphere. The fraction between these two terms is directly driven by the absorptivity of the atmosphere [Eq. (8a)]. When the absorptivity is zero, the total flux leaving the atmosphere originates from radiation emitted by the surface, and the atmosphere has no radiative impact. When the absorptivity is close to 1 (i.e., when the total optical thickness of the atmosphere is larger than about 4), the opposite situation happens: the total flux leaving the atmosphere has been emitted by the atmosphere, the surface does not have any direct radiative impact on the flux leaving the atmosphere, and increasing the optical thickness does not have any influence on the ratio between these two terms anymore. However, this does not mean that the greenhouse effect does not change. Increasing the optical thickness increases the mean emission height and if the atmosphere is not isothermal, a change in emission height translates in a change in outgoing radiative flux.
For an increase in CO2 concentration above its preindustrial value, the increase of the greenhouse effect is primarily due (by about 90%) to the change in emission height. In spectral regions that actually contribute to the radiative forcing, the increase in emission height is about 1 km for a doubling of the CO2 concentration. As the mean emission height is about 6 km (i.e., above where most of the mass of water vapor is located), the radiative effect of this change of emission height is weakly affected by the water vapor amount. This explains why the increase of the greenhouse effect when CO2 increases is weakly dependent on the H2O amount (Fig. 1b), in contrast with the broadband absorptivity. The change in emission height will be of comparable magnitude for any other well-mixed absorbing gases in the spectral domains where the absorptivity is saturated. For an increase of water vapor, the change in absorptivity plays a more important role (about 40%) but the change in emission height is still about 60%. Indeed, away from the atmospheric window, the absorptivity by water vapor becomes saturated and the change in emission height becomes therefore dominant.
The emission height depends on both the temperature profile and the optical properties [Eqs. (14) and (11)]. We showed that the classical assumption that the emission height is close to the altitude where the optical thickness between this altitude and the top of the atmosphere is equal to one is valid only for atmospheres that are optically thick enough (τs > 4). For optically thin atmospheres or with and optical thickness close to one, this assumption is not valid and leads to an underestimation of the emission height.
Considering the real temperature vertical profile in the whole atmosphere makes simplified analysis of the greenhouse effect a priori difficult. However, this complexity is essentially eliminated when considering the adjustment of the stratospheric temperature. This had long been shown when considering global fluxes. Here, we have shown that this is also the case when looking at the change in spectral fluxes and emission altitude, and therefore that it is legitimate to replace the vertical profile of the entire atmosphere by the vertical profile of the troposphere alone, for simplified thinking.
Acknowledgments
We thank the reviewers and the editor for their comments and the many valuable suggestions they made that helped to improve the paper, and we thank Audine Laurian for helping to edit the English. This research was initiated during the internships of Mélodie Trolliet, Thomas Gossot, and Cindy Vida. This work was partially supported by the European FP7 IS-ENES2 project (Grant 312979) and the French ANR project MCG-Rad (18-CE46-0012-03). The Kspectrum and HR_PPart radiative codes are available online (https://www.meso-star.com/en/) under entry “Atmospheric Radiative Transfer.”
APPENDIX A
Analytical Expression for the Idealized Atmosphere
In this section we take advantage of the assumption of the idealized atmosphere (section 2) and in particular that the spectral Planck function increases linearly with pressure [Eq. (4)].
a. Outgoing radiance at the tropopause
b. Weight ω(P)
If the atmosphere is optically thick, τs = κ(Ps − Pt) ≫ 1,
APPENDIX B
REFERENCES
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A second-order Taylor development is required for