Temperature-Dependence of the Clear-Sky Feedback in Radiative-Convective Equilibrium

We quantify the temperature-dependence of the clear-sky climate sensitivity in a one-dimensional radiative-convective equilibrium model. The atmosphere is adjusted to fixed surface temperatures between 280 and 330 K while preserving other boundary conditions in particular the relative humidity and the CO 2 concentration. We show that an out-of-bounds usage of the radiation scheme rapid radiative transfer model for GCMs

2 of 10 (e.g., Romps, 2020) and in doing so shows how λ effectively depends on CO 2 . However, their mechanism fails to explain a similar "bump" in  as temperature increases in constant-CO 2 simulations as in Meraner et al. (2013). Moreover, coupling temperature changes to CO 2 , while physical, makes it difficult to separate the state-dependence of λ on T s from its dependence on CO 2 .
In this study, we calculate  as a function of a fixed T s , for T s ∈ [280 K, 330 K]. After the atmosphere has equilibrated to the boundary conditions and the chosen T s , the radiative feedback is computed as the change in OLR between simulations at increasing T s (Section 2). Calculations were initially performed using a fast radiative transfer model (Mlawer et al., 1997), identical to that used in many climate modeling studies. To check the calculations of the more parameterized fast radiative transfer model, and to understand how the spectral forcing and feedback associated with a doubling of atmospheric CO 2 depends on temperature, we also perform calculations with a line-by-line model. We find that qualitative errors from the fast radiative model become pronounced as T s increases above 300 K, and it overestimates the temperature-dependence of  by more than a factor of two as compared to the line-by-line model reference (Section 3).
Studies of the clear-sky feedback date back to Simpson (1928), who proposed that-in an atmosphere whose optical properties arise from a condensible species (water)-OLR decouples from T s when the atmosphere becomes optically thick. Ingram (2010) brought these ideas to the attention of the climate community (in the meantime planetary scientists, initially unaware of Simpson's work, had come to similar conclusions) and concluded that if the water vapor concentration is a function of temperature only, a warming atmosphere will increases its optical thickness (and hence its emission height) in a way to maintain a constant emission temperature. For Earth's atmosphere this happens when T s > 300 K (Goldblatt et al., 2013;Koll & Cronin, 2018). This decoupling was later (and independently) shown to underpin a limit to how much energy Earth's troposphere can radiate to space in the thermal infrared (Nakajima et al., 1992), with runaway (greenhouse) warming ensuing when the absorbed insolation exceeds this limit (Goldblatt et al., 2013(Goldblatt et al., , 2017Kasting, 1988;Nakajima et al., 1992). These findings encourage the expectation that λ and hence  will increase monotonically with T s , increasingly so for T s > 310 K, rather than to first increase and then decrease, as found by Meraner et al. (2013). Our line-by-line calculations support the conceptual idea of an increasing feedback: beyond a small local plateau, λ continues to increase monotonically with warming. Using the spectral information of our line-by-line simulations we show that this is not only driven by the rapid closing of the atmospheric emission window, but also by water-vapor becoming the dominant absorption species at the flanks of the CO 2 bands. By considering the spectral response to warming and forcing (Section 4) we are able to understand this behavior, also in light of the earlier literature.

Methods and Data
To analyze how the clear-sky climate sensitivity,  , varies with surface temperature, T s , we use the one-dimensional radiative convective equilibrium (RCE) model konrad Kluft et al., 2019). The representation of the climate system, or even just the climate of the tropics, in terms of cloud (as well as aerosol and ozone) free RCE is a strong, but common, simplification. The Charney et al. (1979) report took it as a starting point and a large body of literature since then has found RCE solutions to be informative of how different physical processes influence climate sensitivity. For this reason RCE remains a well studied model problem Goldblatt et al., 2013;Koll & Cronin, 2018;Popke et al., 2013;Seeley & Jeevanjee, 2021;Stevens & Bony, 2013;Wing et al., 2017), one which for reasons elegantly articulated by Polya (1962), is worth first understanding.
Konrad is equilibrated at prescribed values of T s between 280 and 330 K with a fixed relative humidity RH = 80% (see Appendix for a more detailed model description). In contrast to earlier studies (Goldblatt et al., 2013;Koll & Cronin, 2018) we also performed simulations in which the temperature profile and heating rates are allowed to interact. This allows the tropopause temperature to evolve based on the radiative heating instead of setting it to a prescribed value. Although this has a quantitative effect on the feedback parameter, our simulations showed that the simpler approach, that those studies adopted, adequately captured the behavior in the more computationally demanding calculations. For this reason, we focus our attention on results developed using a fixed tropopause temperature of 175 K and an isothermal stratosphere. Figure 1a shows the resulting temperature profiles as a function of atmospheric pressure p.

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The chosen relative humidity of 80% results in a moister troposphere compared to observations in the tropics, which often show a drying especially in the mid-troposphere, i.e., a C-shaped RH profile (Romps, 2014). As the emission window closes, changes in the strength of λ with warming are associated with water vapor progressively stealing ground on the flanks of the CO 2 lines (see Section 4). This behavior, because it depends on the overlap between the CO 2 and H 2 O absorption, is also influenced by the chosen shape of the humidity profile , but not in ways that fundamentally influence our conclusions (see Supporting Information S1). In addition, some assumptions of the 1D-RCE framework presumably become less valid in extreme climates at T s ≫ 310. However, a detailed assessment of the general validity of the RCE framework is beyond the scope of this study, which is why we consider the same temperature range as existing studies (e.g., Meraner et al., 2013;Romps, 2020).
The RCE simulations are performed for CO 2 concentrations of 348 and 696 ppmv which allows us to compute the radiative forcing ΔF and the feedback parameter λ. We define ΔF at a given T s as the difference in net radiation balance ΔN at the top of the atmosphere between these two CO 2 concentrations The feedback parameter λ is defined as the change in ΔN between simulations at constant CO 2 = 348 ppmv and different T s with surface temperature difference ΔT = 1 K. Following our definition, a negative λ is associated with a stabilizing effect on the climate system. With this approach, we can study the temperature-dependence of the radiative forcing ΔF, the climate feedback λ, and the resulting climate sensitivity  = −Δ ∕ . Note, that our definition of  does not account for changes in λ due to the doubling of the CO 2 concentration, which is usually included in estimates of  .
To check our method we have also performed simulations with a coupled T s and computed λ as the regression of ΔN over ΔT s during a perturbed simulation (Gregory et al., 2004). We find that the results are in very good agreement with those obtained using Equation 2. However, the strong temperature-dependence of λ makes the linear regression error-prone, which mostly manifests itself in spurious signals in the estimated effective forcing (y-intercept of the regression). Therefore, we opted for the well-established fixed-T s approach.

Figure 1.
Equilibrium temperature (a) and water-vapor volume mixing ratio (b) profiles at different surface temperatures but constant CO 2 concentrations as a function of atmospheric pressure. The figure is clipped at 5 hPa to better visualize the troposphere. In addition, the rapid radiative transfer model for GCMs (RRTMG) reference temperature range, and the maximum (pressure) height up to which RRTMG considers the water-vapor continuum are shown.

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Baseline simulations are performed using the Rapid Radiative Transfer Model for GCMs (RRTMG) (Mlawer et al., 1997). RRTMG is a fast radiation scheme which uses the correlated-k method with precalculated lookup tables for computational efficiency. For line-by-line simulations we replace the RRTMG longwave radiative transfer calculations with calculations using the Atmospheric Radiative Transfer Simulator (ARTS) (Buehler et al., 2018;Eriksson et al., 2011) which has been part of the Radiative Forcing Model Intercomparison Project (RFMIP, Pincus et al., 2020). In the chosen setup, ARTS represents the longwave radiative fluxes based on 32768 equidistant frequency points between 10 cm −1 and 3250 cm −1 (Δν = 0.1 cm −1 ). Explicitly resolving the spectrum of OLR later allows us to investigate conceptual ideas about the dependence of OLR on T s in different spectral regions.
For the sake of simplicity and to facilitate comparisons with previous modeling studies we do not consider the effects of ozone. We have performed calculations in which ozone is allowed to change, and while the basic physics that we describe are not influenced by this elaboration, as λ becomes small the effect of ozone can become important. Quantitatively its influence is found to depend on the details of its representation, particularly in light of the deepening of the troposphere with warming, an interesting issue that we are beginning to explore together with experts on ozone chemistry.
Further information about konrad's configuration, RRTMG, and ARTS is given in the Appendix.

Temperature-Dependence of the Feedback Parameter λ
We run konrad for T s between 280 and 330 K to quantify the temperature-dependence of the feedback parameter with constant values of insolation I, surface albedo α, relative humidity RH, and the gaseous composition χ. Seeley and Jeevanjee (2021) consider the related problem λ = f(T s , CO 2 ; …), where the CO 2 concentration is variable.
For low temperatures, calculations based on RRTMG and ARTS agree well with one another. Figure 2 (solid lines) shows the radiative forcing ΔF and the feedback parameter λ (as defined in Section 2), as well as the resulting equilibrium climate sensitivity  , as a function of T s . Both results, using either RRTMG (gray) or the line-by-line radiative transfer model ARTS (green), show that λ (Figure 2b) increases from −2.1 to −1.3 W m −2 K −1 as T s increases from 280 to 300 K. A more detailed feedback analysis (not shown) identifies this increase with the temperature-dependence of the water-vapor feedback.
For T s > 300 K, calculations with RRTMG result in a pronounced local maximum, or "bump", in  . This is seen in Figure 2c, where  increases from less than 3 K at T s = 300 K to about 10 K at T s = 310 K, and then rapidly decreases to less than 2 K at T s = 320 K. Figure 2 further shows that RRTMG's response can

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be attributed to changes of the feedback parameter λ, rather than the forcing. Hence the bump, and its origins, are similar to what was found in other studies (Meraner et al., 2013;Romps, 2020) using correlated-k radiative transfer. When using ARTS, however, λ does not decrease for T s > 305 K. In contrast, after a local plateau around 305 K, λ begins to increase again at higher T s . This is consistent with work by Goldblatt et al., 2013 (their Figure 4) who show that, at even higher T s , water-vapor controls the emission in the whole OLR spectrum provoking a runaway greenhouse. Neither their study nor our results support the existence of a "bump" in clear-sky λ.
In addition, we performed another set of experiments with increased CO 2 concentrations (dashed lines in Figure 2) following the experimental setup by Romps (2020) and Seeley and Jeevanjee (2021). Using ARTS, we confirm the stabilizing effect of increased CO 2 concentrations as explained by the "radiator fins" described in Seeley and Jeevanjee (2021). However, simulations using RRTMG show the same qualitative behavior irrespective of the chosen CO 2 concentration. This strengthens our interpretation that the increase in λ as predicted by RRTMG has no physical explanation but is caused by inaccuracies in the treatment of radiative transfer.
RRTMG, and other fast-radiative transfer schemes, aggregate absorption features into bands, within which optical properties are calculated by interpolating across pre-computed look-up tables. This reduces the computational intensity and speeds up the calculations. In RRTMG the lookup tables are based on an assumed atmospheric composition and thermal structure, close to those of the present-day Earth (Mlawer et al., 1997, their Section 3.2). As it turns out, how one interprets the word "close" can be problematic. For instance, while RRTMG is documented to be valid for T s as high as 320 K, this is based on a temperature profile representative of mid-latitude summer and assumes that there are no changes in the temperature lapse rate with increasing surface temperature (Mlawer et al., 1997, their Section 3.2). As a consequence, the temperature lapse-rate in the lookup table is larger than the moist-adiabat, which implies mid-and upper-tropospheric temperatures that are out of bounds at T s above 306 K (see Figure 1a). Popp et al. (2015) attempted to minimize the resultant errors by extending the temperatures to acceptable bounds when performing the gaseous look-up. The look-up tables are only one source of error. Another, which we identified, arises from RRTMG's calculation of the water-vapor self continuum. For computational expediency this is fit to only two reference temperature values (at 260 and 296 K) and is neglected entirely for pressures less than 100 hPa. The latter becomes increasingly error prone as T s increases above 296 K and the deepening troposphere causes a moistening above 100 hPa (see Figure 1b). For the case of RRTMG, these errors lead to an overestimation of OLR, which is misinterpreted as a decrease (more negative) of λ at high T s . Coincidentally, this happens around the same temperature range at which the CO 2 mechanism described by Seeley and Jeevanjee (2021) begins to work.
In conclusion, using a line-by-line radiation model we find a robust increase of λ for T s up to 330 K. Errors in the calculation of longwave irradiances by RRTMG are shown to be the cause of a spurious "bump" in clear-sky λ. This "bump" looks similar, but is entirely unrelated, to the local maximum in λ that Seeley and Jeevanjee (2021) find (and physically explain), when CO 2 is allowed to covary with T s . For fixed CO 2 , as T s increases, λ increases, but remains more negative than −1 W m −2 K −1 even for temperatures as high as 330 K. At yet higher temperatures, and as reported by Goldblatt et al. (2013) for temperatures around 350 K, this negative feedback might completely vanish, albeit for a setup that is becoming increasingly artificial. Its substantially negative value for T s as high as 330 K was less expected, something we address in more detail in the following section.

Spectral Analysis of λ and ΔF
To understand why λ is far from zero even at T s = 330 K, we here examine the spectral feedback parameter λ ν . This framework was used by Kluft et al. (2019) as well as Seeley and Jeevanjee (2021), and can also be used to study the role of different spectral regions in changing ΔF and  . The important difference between our situation, and the situation envisioned by Simpson (1928), is that H 2 O is not the only absorber in the infrared. Were that the case it would not be possible to force the system by increasing atmospheric concentrations of CO 2 . The problem as we pose it here, is not how Earth can respond to energy accumulated by an external process, such as insolation or accretion of extra-planetary material (Abe & Matsui, 1988; 6 of 10 Kasting, 1988;Nakajima et al., 1992), but rather how the reduction of infrared irradiance of the atmosphere can be compensated through warming.
The spectral feedback parameter λ ν can be derived from our line-by-line calculations using Equation 2. Figure 3b shows the smoothed λ ν as a function of wavenumber ν for simulations at different temperatures (and hence absolute humidity). There is a strong temperature-dependence of λ ν in the atmospheric window between 715 and 1,250 cm −1 . This is driven by the increasing water vapor concentration in the warming troposphere, as λ ν is indeed close to zero as soon as the atmosphere becomes fully opaque at high temperatures (darker blue shades) and stays close to zero for higher T s . Hence, our results link the findings of Koll and Cronin (2018) with the studies by Nakajima et al. (1992) and Goldblatt et al. (2013).
In our simulations the total λ remains negative for all T s up to 330 K. We attribute this mainly to the thermal Planck feedback in the CO 2 bands around 667 cm −1 . Adopting the analogy introduced by Seeley and Jeevanjee (2021), the infrared emission attributable to tropospheric CO 2 acts as a spectral radiator fin, stabilizing the climate to greenhouse forcing. This effect, however, is limited by the water-vapor amount in the atmosphere: with increasing T s , water-vapor controls an ever growing part of the emission spectrum. The deepening of the troposphere raises the emission level to lower pressure thereby reducing the impact of the CO 2 band (see Figure S3 in Supporting Information S1). Eventually, at T s even higher than simulated in our study, it sets the emission temperature of the whole OLR resulting in a zero feedback. The value of T s , at which this runaway greenhouse state is reached, depends on the relative humidity of the atmosphere. A subsaturated atmosphere column is stable at higher T s (see Figure S1 in Supporting Information S1), which explains why our feedback is stable for even higher T s than for the fully saturated simulations by Goldblatt et al. (2013).
Similar to the increase of λ we observe a reduction in ΔF with warming. Usually, the radiative forcing is thought to increase with T s (Huang et al., 2016). Such an effect is apparent in our simulations, but only for lower values of T s , up to 300 K (Figure 2a). This strengthening of ΔF with warming arises from a larger contribution from the band center (between 620 and 700 cm −1 ). At higher T s , ΔF decreases, so that with T s = 320 K it is 18% less than its value at 295 K. The reduction in ΔF with warming is due to a weakening KLUFT ET AL.
10.1029/2021GL094649 7 of 10 contribution from the edges of the 667 cm −1 CO 2 band (see Figure S2 in Supporting Information S1). At T s = 280 K, 15% of the forcing is carried by the band center, at 320 K the forcing from the band center has increased more than threefold and is responsible for 60% of the total forcing. CO 2 absorption is so strong near the central absorption feature, that emission to space from these wavelengths originates in the stratosphere. Only lines whose emission height resides in the troposphere, where temperatures decrease with height, contribute to reduced emissions-and hence forcing-from increasing CO 2 concentrations. As the tropopause rises with warming, an increasing fraction of the OLR originates from CO 2 in the troposphere, and its changes can contribute to the forcing. As increasing water vapor closes the window at T s > 300 K, emission by H 2 O increasingly dominates over emission by CO 2 on the flanks of the CO 2 band. This reduces the contribution of tropospheric CO 2 to the OLR, thereby reducing the contribution of its changes to forcing. The latter increasingly dominates at warmer temperatures, weakening ΔF from a doubling of CO 2 by about 18% (from a value around 4.5 to 3.7 Wm −2 for T s increasing from 280 to 320 K), consistent with an analytical model of the CO 2 forcing by Jeevanjee et al. (2020). Our analysis identifies the same mechanism-the increasing dominance of water-vapor over the OLR-to affect both λ and ΔF in opposite ways, albeit with very different impact. Seeley and Jeevanjee (2021) demonstrated how an increase in CO 2 concentration strengthens the CO 2 absorption band in the atmospheric window: at some point the CO 2 replaces H 2 O as the dominant absorber and acts as a "CO 2 radiator fin". To understand the effects of warming on both λ and ΔF we find a different analogy helpful. We picture a "CO 2 archipelago in a developing, and eventually rising, sea of water-vapor absorption" (see Figure 4, the poetically inclined might think of these as Planckian outcroppings in a Simpsonian sea). From this point of view the share of the radiation that is emitted to space by H 2 O in the troposphere, versus that from tropospheric CO 2 or from the surface, determines the strength of both λ and ΔF. As CO 2 concentrations rise, or the troposphere deepens, the CO 2 archipelago gains prominence-new islands even appear with rising CO 2 concentrations, as seen in Seeley and Jeevanjee (2021)-increasing the magnitude of both ΔF and λ. Warming of the atmosphere leads to the development of a "sea of absorption," which progressively reclaims the spectral landscape from CO 2 and the surface. This reduces ΔF for a given increase in CO 2 and progressively masks the ability of radiation from the "sea-floor" to escape to space. In our simulations, at T s = 320 K the "absorption sea-level" is so completely determined by temperature, as envisaged by Simpson (1928), that the net radiative response to warming in the window region, 715 cm −1 < ν < 1250 cm −1 , completely vanishes. At this point only the tallest mountains of the "CO 2 archipelago", whose prominence is pronounced due to a rising tropopause, are left to balance an increase in forcing. The left panel shows how the "CO 2 archipelagos" grow with an increase in atmospheric CO 2 concentration (darker greys). The right panel shows the rising "H 2 O absorption sea-level" at higher water vapor volume mixing ratios (darker blues, the label states surface VMR while the actual humidity profile is computed based on a constant RH of 80%). In addition, the τ = 1 line roughly indicates the location of opaque spectral regions (τ > 1).

Discussion and Conclusions
We perform calculations using the 1D-RCE model konrad at different surface temperatures T s to analyze the temperature-dependence of the feedback parameter λ for fixed and variable CO 2 concentrations. A lineby-line treatment of longwave radiant energy transfer (ARTS) is used to ensure an accurate computation of radiative fluxes and heating rates over a wide temperature range. By comparison to calculations with the RRTMG radiation scheme, we find that the use of the latter (albeit faster) scheme leads to increasingly erroneous results as surface temperatures increase beyond 300 K-errors in climate sensitivity are larger than a factor of two at 310 K. This is within the range of temperatures sampled by models with very high climate sensitivities subject to quadrupling of atmospheric CO 2 . The erroneous behavior leads to a local maximum (or "bump") in the clear-sky climate sensitivity. Coincidentally, a similar "bump" is predicted by the "CO 2 radiator fin" mechanism by Seeley and Jeevanjee (2021), which arises from the strengthening of the Planck feedback from more pronounced CO 2 absorption features. Using two different sets of simulations, we show that RRTMG computes the "bump" irrespective of the CO 2 concentration. The resulting unphysical trend in climate sensitivity looks similar to what has been found in at least two other modeling studies (e.g., Meraner et al., 2013;Romps, 2020) using this same, or a similar, treatment of radiative transfer. In Romps (2020) both effects, the large increases in CO 2 , which the climate sensitivity also depends on, and the RRTMG errors are conflated. However, our simulations suggest that the erroneous treatment of radiative transfer alone is sufficient to cause the reversing trend in climate sensitivity.
Using ARTS and a constant CO 2 concentration, λ increases from −2.1 to −1.0 W m −2 K −1 for T s between