## 1. Introduction

How fast would Earth’s climate respond to increasing CO_{2} (Manabe and Wetherald 1975; Flato et al. 2013; Collins et al. 2013)? Why is tropical climate more stable than extratropical climate (Holland and Bitz 2003; Polyakov et al. 2002; Pierrehumbert 1995)? What sets the inner edge of the habitable zone of Earth-like planets (Yang and Abbot 2014; Pierrehumbert 2010a)? Understanding and accurately estimating climate feedbacks are key to address these pressing questions.

The importance of water vapor seems to be widely recognized in the literature of climate feedbacks (Manabe and Wetherald 1967; Ingersoll 1969; Held and Soden 2000; Flato et al. 2013). Previous studies have focused on three basic effects of water vapor: E1) water vapor is a greenhouse gas; E2) water vapor can condense to liquid water and release latent heat; and E3) saturation vapor pressure increases with temperature exponentially. The combination of E1 and E3 gives rise to the water vapor feedback, the dominant positive climate feedback (Manabe and Wetherald 1967; Held and Soden 2000; Flato et al. 2013). Increasing temperature leads to more water vapor, which leads to an enhanced greenhouse effect, warming the planet further. The water vapor feedback could even lead to a runaway greenhouse state when the atmosphere is sufficiently opaque to longwave radiation that the outgoing longwave radiation (OLR) is insensitive to surface temperature (Ingersoll 1969). The combination of E2 and E3 gives rise to the (tropical) lapse rate feedback, a negative climate feedback in the tropical atmosphere (Flato et al. 2013). Increasing temperature leads to more water vapor, which leads to a weaker lapse rate in the tropical atmosphere. This effect increases upper troposphere temperature more than in the lower troposphere, leading to higher emission of OLR, which cools the planet. At higher latitudes, the temperature lapse rate is no longer controlled by moist convection, so the lapse rate feedback is less constrained. Both feedbacks are among the five most important climate feedbacks in the Intergovernmental Panel on Climate Change (IPCC) reports and have been extensively evaluated in general circulation models (GCMs) (Flato et al. 2013).

*T*is temperature,

*r*is water vapor mixing ratio, and

*ε*=

*M*

_{υ}/

*M*

_{d}. The molar mass of water vapor

*M*

_{υ}is 18 g mol

^{−1}, significantly lighter than that of dry air

*M*

_{d}, which is 29 g mol

^{−1}. This makes a moist parcel lighter than a dry parcel of the same temperature and pressure (Emanuel 1994). Here we refer to this as the vapor buoyancy effect, although it is also referred to as the virtual effect (Yang 2018a,b). The vapor buoyancy effect makes

*T*

_{υ}slightly higher than

*T*by

*O*(1 K) in the tropical atmosphere, which is of the same order of magnitude as typical temperature fluctuations there.

We propose that the vapor buoyancy effect can increase Earth’s OLR and help to stabilize Earth’s climate by regulating the atmosphere’s thermal structure. Figure 1 shows the temperature and virtual temperature (buoyancy) fields in the moisture space from 2°S to 2°N using NASA AIRS data. In the free troposphere (*p* < 850 hPa), buoyancy is horizontally uniform because of the small Coriolis parameter and efficient gravity waves (Charney 1963; Bretherton and Smolarkiewicz 1989; Sobel et al. 2001; Yang 2018a). However, temperature increases toward the dry columns as a result of the vapor buoyancy effect. There, moisture and its associated vapor buoyancy are reduced. To maintain uniform buoyancy, temperature has to increase. We propose that the temperature tilt would increase with climate warming due to increasing atmospheric moisture, leading to enhanced OLR over the dry area. This is a negative feedback that can help to stabilize Earth’s climate.

Previous studies implied that vapor buoyancy could make temperature increase toward the dry columns in the tropical atmosphere (Tompkins 2001; Bretherton and Smolarkiewicz 1989; Bretherton et al. 2005; Yang 2018a,b). However, they have often considered this effect to be small and negligible, simplifying the dynamics according to a weak temperature gradient approximation (Sobel et al. 2001). These studies, therefore, did not consider that its radiative effect is significant, which is the novelty of this study.

In section 2, we explain our hypothesis in detail. We first illustrate how the vapor buoyancy effect increases Earth’s OLR (a negative radiative effect) and then explain why this effect strengthens with climate warming. In section 3, we derive a simple model for the radiative effect and feedback strength of the vapor buoyancy effect. We then use the simple model to make order-of-magnitude estimates for the radiative effect and feedback strength. In section 4, we estimate the radiative effect by using tropical sounding profiles with a real-gas radiative transfer model. In section 5, we conclude and discuss implications for the climate stability of Earth and other planets.

## 2. Hypothesis

We propose that vapor buoyancy can increase OLR (a negative radiative effect) due to a clear-sky effect, and that the radiative effect increases with climate warming. Figure 2 illustrates our hypothesis by comparing OLR from two stand-alone atmospheres with overturning circulations: one considers the vapor buoyancy effect (control) and the other does not consider this effect. The overturning circulation is analogous to the Walker circulation or convective self-aggregation in the tropics (Bretherton et al. 2005; Pritchard and Yang 2016; Yang and Ingersoll 2013, 2014; Yang 2019). The upwelling branch of the circulation is associated with deep convection and moist air, and the downwelling branch is associated with clear skies and dry air. For illustrative purposes, we make a few simplifications: S1) the two atmospheres are nonrotating, S2) the two atmospheres sit above ocean surface with the same, uniform surface temperature, and S3) the two atmospheres have the same water vapor distribution. The first two simplifications are relevant to the tropical atmosphere as the rotation effect and surface temperature gradient are both weak in the tropics. The third simplification is often required when calculating a radiative effect.

Figure 2 shows that the control atmosphere emits more OLR than the no-vapor-buoyancy atmosphere due to the higher temperature in the dry area. OLR is primarily a function of temperature and water vapor mixing ratio *r*. When *r* remains the same in the two atmospheres (S3), the OLR difference would come from temperature differences between the two atmospheres. Here we provide physical intuition on why there should be temperature differences, leaving a detailed derivation for section 3e. The temperature profiles of moist areas in the two atmospheres are set by convective plumes. Because these convective plumes rise from the same surface temperature, the temperature profiles should be almost identical in the two moist areas. Temperature profiles in the dry areas, however, differ significantly, leading to differences in OLR. According to long-accepted results in geophysical fluid dynamics, the horizontal buoyancy gradient is negligible in the free troposphere without rotation because gravity waves can effectively smooth out buoyancy anomalies (Charney 1963; Sobel et al. 2001). We refer to this effect as the weak buoyancy gradient (WBG) approximation (Yang 2018a). In the control atmosphere, buoyancy is a function of both temperature and and water vapor mixing ratio *r* due to the vapor buoyancy effect. The horizontal moisture gradient then leads to a horizontal temperature gradient: dry air is warmer than moist air. In the no-vapor-buoyancy atmosphere, the free-troposphere temperature is horizontally invariant, as buoyancy is a function of temperature only. The dry column of the control atmosphere, therefore, is warmer than that of the no-vapor-buoyancy atmosphere by *O*(1 K), leading to enhanced OLR. The spectrum of H_{2}O in the longwave is also sensitive to temperature. However, this impact is likely small.

In warmer climates, the vapor buoyancy effect would become more significant due to increasing water vapor. Therefore, we expect that the radiative effect due to the vapor buoyancy also increases with climate warming. This is a negative climate feedback (Fig. 2b). The proposed mechanism relies on ample atmospheric water vapor, so it would be most effective in stabilizing the tropical climate. In principle, this feedback should have been represented by climate models. However, it has not been evaluated or even discussed.

We will construct a simple model of the proposed feedback mechanism. This will give an order-of-magnitude estimate of the associated radiative effect and the rate at which it increases with climate warming.

## 3. A simple model

We construct a simple model based on the schematic diagram (Fig. 2). Each atmosphere with overturning circulations is represented by a dry column and a moist column (Pierrehumbert 1995). Because the the two atmospheres’ respective moist columns would have the same temperature profiles, the OLR difference primarily comes from their dry columns, which we will focus on. Again, we aim to estimate the “radiative effect” due to the vapor buoyancy effect. Therefore, we assume that all basic dynamic (e.g., circulation and pressure) and thermodynamic features (e.g., moisture) are the same in the two atmospheres—one with the vapor buoyancy effect, and the other without it.

The goal of this simple model is to provide an order-of-magnitude understanding of our hypothesis. Therefore, we employ a two-band radiative transfer model. The two-band model is more realistic than a gray atmosphere model by allowing two absorption bands with distinct absorption coefficients, leading to different emission levels. The two-band model is, on the other hand, much simpler than a real-gas radiative transfer model, so the results are easier to interpret. To further test our hypothesis and the validity of the simple model, we also show calculation results using a real-gas radiative transfer model in the appendix. The results are in good agreement between the two models, suggesting that our hypothesis is based on robust physics and not on model details.

### a. The two-band model

We consider a plane-parallel atmosphere and use the two-stream approximation in the following calculations. Only clear-sky longwave (IR) radiation is considered, and the IR opacity is mainly due to water vapor. Here we parameterize the water vapor absorption spectrum by two broad bands that occupy roughly equal fractions of blackbody emission at Earth-like temperatures (Beucler and Cronin 2016): one with a strong absorption coefficient *κ*_{S} and the other with a weak absorption coefficient *κ*_{W}.

*κ*. OLR is defined as

*F*

^{↑}and

*F*

^{↓}are upward and downward longwave radiative fluxes (Pierrehumbert 2010b). We know that

*F*

^{↓}(0) ≈ 0, so we shall solve for

*F*

^{↑}(0) in the gray atmosphere, which is given by

*T*is temperature,

*τ*is optical depth, and

*σ*is the Stefan–Boltzmann constant. This is the two-stream Schwarzschild equation (Pierrehumbert 2010b). We integrate Eq. (3) and get

*A*

_{s}represents the surface value of some variable

*A*, and we have used

*T*

_{s},

*r*, and thereby

*τ*of the two atmospheres are identical, so the OLR difference primarily comes from dry columns, between which there is significant difference in air temperature. The OLR difference of the dry column is given by

_{υ}and OLR

_{nv}represent OLR in the atmosphere with and without the vapor buoyancy effect. In the last equality, we assumed that (

*T*

_{d}−

*T*

_{m})/

*T*

_{m}≪ 1. Because the strong and weak absorption bands occupy equal portions of the spectrum, the total OLR difference is given by

*T*

_{m}, Δ

*T*,

*τ*, and thereby

*r*, which is the mixing ratio of water vapor.

### b. Temperature

*T*

_{s}is the surface temperature,

*p*

_{s}is surface pressure,

*R*

_{d}is the gas constant for dry air, Γ

_{M}is the moist adiabatic lapse rate, and

*g*is gravity acceleration. This has been referred to as the “all-troposphere model” by Pierrehumbert (2010b), as the lapse rate is entirely determined by moist convection. Equation (8) fits the observed temperature profiles in the tropical troposphere, but introduces significant biases in the stratosphere (Beucler and Cronin 2016). Earth’s OLR is dominated by tropospheric contributions, which justifies the use of Eq. (8).

### c. Moisture

*r*is the ratio of the mass of water vapor to the mass of dry air and is given by

*r*

^{*}is the saturation mixing ratio. For the moist column, we assume that

*β*< 1. Here

*β*is a more convenient parameter than the relative humidity of dry columns (RH

_{d}). This is because, at given

*T*

_{s}, the two atmospheres’ dry columns are of different temperatures, so they would have different RH values corresponding to the same mixing ratio. In reality,

*β*could have complicated vertical structures, which requires multiple parameters to describe. For the purpose of illustrating the proposed mechanism with minimal parameters, we take

*β*as a constant at all vertical levels. This simplification may affect the results quantitatively but will not affect the results qualitatively.

### d. Optical depth

*τ*as an increasing function as pressure. We thus require that

*τ*(

*p*= 0) = 0. We write the optical depth as

*dτ*=

*κ*

^{s}

*rdp*/

*g*for the strong band, and

*dτ*=

*κ*

^{w}

*rdp*/

*g*for the weak band. In this model, we ignore the pressure-broadening effect and treat the absorption coefficients as constant:

*κ*

^{s}= 1.66 × 0.1 (m

^{2}kg

^{−1}) and

*κ*

^{w}= 1.66 × 0.02 (m

^{2}kg

^{−1}), where the factor of 1.66 is referred to as the diffusivity factor (Pierrehumbert 2010b). Here the absorption coefficients are consistent with previous modeling studies of similiar complexity (Ingersoll 1969; Pierrehumbert 2010b; Beucler and Cronin 2016). We can integrate Eq. (10) to obtain the optical depth at an arbitrary pressure level:

*p*=

*p*

_{s}, we then have

*τ*

_{s}measures the optical depth at the surface, which is total optical depth of the atmospheric column.

### e. The WBG approximation and *Δ*T

*ε*=

*M*

_{υ}/

*M*

_{d}, with

*M*

_{υ}and

*M*

_{d}representing the molar mass of water vapor and dry air, respectively. In the free troposphere, uniform buoyancy requires the virtual temperature to be uniform across the moist and dry area:

*T*

_{d}=

*T*

_{m}+ Δ

*T*

_{WBG}into Eq. (13) and get

*T*. Although this form is very accurate, we would like to simplify it by assuming water vapor is a trace gas:

*r*≪ 1. This is a good assumption for the current climate and may still be good until surface temperature reaches 320 K, at which temperature

*r*

^{*}(

*p*

_{s}) = 73 g kg

^{−1}. With this approximation, we get

*T*depends on the contrast, not just absolute values, of mixing ratio and molar mass.

*T*= 0 at the surface because of the uniform sea surface temperature (SST). Therefore, we require Δ

*T*to be equal to Δ

*T*

_{WBG}in the free troposphere but to smoothly decay to 0 at surface:

*n*controls the decay rate with pressure. The

*p*/

*p*

_{s}term would decay faster with large

*n*and slower with small

*n*, so different

*n*could potentially result in different amplitudes and altitudes of the maximum temperature difference. However, we find that the values of ΔOLR and its sensitivity to surface temperature only change by 50% while we vary

*n*over an order of magnitude, from 5 to 50. Therefore, we conclude the results are robust to the choice of

*n*, and we take

*n*= 30 in the following calculation. Figure 3a shows Δ

*T*profiles with different

*β*values at different surface temperatures: Δ

*T*decreases with

*β*and increases with

*T*

_{s}. At 300-K surface temperature, the maximum of Δ

*T*is about 0.45 K when

*β*= 0.8; the maximum of Δ

*T*is about 1.1 K when

*β*= 0.5; and the maximum of Δ

*T*is about 1.7 K when

*β*= 0.2. We find that Δ

*T*increases faster with

*T*

_{s}in drier columns, which would be used to explain the sensitivity of ΔOLR to

*T*

_{s}.

Equations (6)–(10) and (16) form the complete set of this model. With proper parameter values, we can estimate the magnitude of ΔOLR and its change with surface temperature.

### f. Results

Our calculation shows that the vapor buoyancy effect can significantly impact Earth’s energy balance and future climate changes. Figure 4a shows that ΔOLR is *O*(4 W m^{−2}) for a wide range of parameter values. In the reference climate (*T*_{s} = 300 K), ΔOLR is about 2.5 W m^{−2} with *β* = 0.5, a similar magnitude to the radiative effect due to doubling CO_{2}. For small values of Δ*T*, as in the present climate, the magnitude of *β* is comparable to that of RH. Therefore, a value of *β* = 0.5 corresponds to a tropical average, while smaller values of *β* might reflect subtropical dry regions [e.g., Fig. 18.3 in Vallis (2017)]. We then understand the sensitivity of ΔOLR to *T*_{s} and *β* according to Eq. (6):

- The ΔOLR increases with
*T*_{s}at given*β.*This is mainly because Δ*T*increase with warming, as will be quantified in Figs. 4b and 4c. - The ΔOLR is small at both moist and dry limits. In the moist limit (
*β*→ 1), Δ*T*is small according to Eq. (15). In the dry limit (*β*→ 0), although Δ*T*maximizes, ΔOLR is dominated by surface emission, insensitive to Δ*T.*The OLR difference, therefore, peaks at intermediate*β*values. - The ΔOLR peak shifts toward smaller
*β*in warmer climates. This is because, at high temperatures, Δ*T*increases faster with warming in the small-*β*columns (Fig. 3a) and also because the large-*β*columns become increasingly opaque to IR emission (Figs. 3b,c).

*T*

_{s}, showing a negative climate feedback. To quantify the feedback strength, we define feedback parameters

*λ*

_{t}is the total sensitivity of ΔOLR to

*T*

_{s}, and

*λ*

_{vb}is the vapor-buoyancy feedback parameter, which only concerns

*d*Δ

*T*/

*dT*

_{s}. That is,

*λ*

_{t}represents the total feedback resulting from adding the vapor buoyancy effect to the atmosphere;

*λ*

_{vb}reflects only the changing temperature difference between the two atmospheres, excluding Δ

*T*’s interaction with other changes in temperature and water vapor. Figure 4b shows that

*λ*

_{t}is of

*O*(0.2 W m

^{−2}K

^{−1}) in the reference climate, which compares with the feedback strength due to clouds and surface albedo. The feedback parameter keeps increasing with surface temperature and reaches about 1.4 W m

^{−2}K

^{−1}at 320 K, suggesting that the vapor buoyancy effect becomes increasingly important in future climates.

Figure 4c shows that *λ*_{vb} is of similar magnitude to *λ*_{t}, suggesting the vapor-buoyancy feedback dominates the entire ΔOLR sensitivity to *T*_{s}. We find that *λ*_{vb} is small at the moist and dry limits. This is because Δ*T* → 0 when *β* → 1 at all surface temperatures, and ΔOLR is dominated by surface emission when *β* → 0 at all surface temperatures, not feeling Δ*T* and its changes. In addition, we find that the peak of *λ*_{vb} moves toward small-*β* columns with warming because Δ*T* increases faster with warming at small-*β* columns (Fig. 3a), and also because large-*β* columns become increasingly opaque at high temperature (Figs. 3b,c), insensitive to changes of Δ*T* that peaks in the lower troposphere.

The overall results do not depend on the assumed Δ*T* profiles in the boundary layer. Figures 4d–f shows ΔOLR, *λ*_{t}, and *λ*_{vb} for the free troposphere (*p* < 900 hPa). The free-troposphere results almost reproduce the full-column results, with amplitudes that are 10%–15% weaker than the full-column calculation. This suggests that the vapor-buoyancy radiative effect and feedbacks occur primarily in the free troposphere.

To further test our hypothesis, we also use a real-gas radiative transfer model to calculate ΔOLR and *λ*_{t}. The results agree well with the two-band radiative transfer calculations. See the appendix for details.

## 4. Observation: A case study

We estimate the radiative effect due to the vapor buoyancy using in situ observed temperature and moisture profiles from the Nauru Atmospheric Radiation Measurement (ARM) site during the period from April 2001 to August 2013 (Figs. 5a–c). The method of this calculation is based on the two-column model developed in Fig. 2 and in section 3. We assume that, in the free troposphere, buoyancy of this atmospheric column is the same as that of a saturated, convecting atmospheric column, and then we solve for the temperature profile of the convecting column using Eq. (13). This calculation requires *T* (Fig. 5c) and *r*, which is based on the observed specific humidity profile (Fig. 5a). This derived temperature profile corresponds to that of an atmosphere without the vapor buoyancy effect (Fig. 2a) and is about 0.8 K colder than the observed temperature profile in the lower troposphere (Fig. 5d), with relative humidity about 0.6 (Fig. 5b).

We then compute the clear-sky OLR of the two columns by using the Rapid Radiative Transfer Model (RRTM v3.3; Mlawer et al. 1997). The model was run using 28 vertical levels between 1013 to 55 hPa. The CO_{2} mixing ratio is set at 400 ppm, and all other trace gases, including ozone, were set to zero. The clear-sky OLR calculated for the observed atmospheric column is 300.4 (W m^{−2}); the clear-sky OLR calculated for the no-vapor-buoyancy column is 299.3 (W m^{−2}). Therefore, the vapor buoyancy effect is responsible for about a 1.1 W m^{−2} increase in clear-sky OLR for a typical atmospheric column in the deep tropics. This result is encouraging because it agrees well with our simple model calculations at 300-K surface temperature. If we shift the boundary layer top from 900 to 800 hPa, this OLR difference would be 0.9 W m^{−2}, which remains significant. This sensitivity test suggests that the radiative effect of vapor buoyancy primarily originates from the free troposphere.

This case study is a first step toward quantifying the radiative effect of vapor buoyancy in Earth’s atmosphere. Future analyses should expand this study by using global-scale datasets, providing a more accurate estimate of the radiative effect of vapor buoyancy over the entire tropical atmosphere.

## 5. Conclusions and discussion

The conventional wisdom is that the vapor buoyancy effect is small, so its impact on temperature is negligible in the free troposphere. However, using NASA AIRS observations, we have demonstrated that the vapor buoyancy effect could lead to about a 1.5-K horizontal temperature difference in the lower troposphere from the driest column to the moistest column (Fig. 1), which has a significant impact on Earth’s radiative balance.

Based on that novel observation, this paper proposes that the vapor buoyancy effect can increase Earth’s OLR by increasing the air temperature in the dry columns. We have developed a simple model that computes the OLR difference between two atmospheres: one with the vapor buoyancy effect, and the other without this effect. We show that the magnitude of this effect is of *O*(1 W m^{−2}) at *T*_{s} = 300 K, which is then confirmed by observations, and that it increases rapidly with climate warming due to an exponential increase of atmospheric water vapor, leading to a negative climate feedback (Fig. 2b). We further show that the feedback strength *λ* is of *O*(0.2 W m^{−2} K^{−1}), the amplitude of which compares with major climate feedbacks, including cloud and surface albedo feedbacks. Therefore, faithful representation of the vapor buoyancy effect in climate models is necessary for accurate estimates of climate sensitivity and reliable predictions for future climate changes. While most GCMs include the vapor buoyancy effect in the model formulation, the magnitude of its associated radiative effect depends on faithfully representing the water vapor distribution in the free troposphere. In future studies, we would like to quantify this uncertainty by estimating the radiative effect of vapor buoyancy in climate models and comparing it with observations.

The vapor buoyancy effect may help explain why tropical climate has been more stable than extratropical climate (Holland and Bitz 2003; Polyakov et al. 2002; Pierrehumbert 1995). The strength of the vapor buoyancy feedback depends on water vapor contrast between moist and dry columns, which in turn depends on water vapor abundance and thereby on the temperature of the atmosphere. This effect, therefore, operates more efficiently in the tropics and less efficiently at higher latitudes. This spatial pattern may explain why fluctuations of sea surface temperature in the tropics are much smaller than that of higher latitudes in the past 100 million years (Pierrehumbert 1995).

The vapor buoyancy effect helps extend the inner edge of the habitable zone, in particular, for tidally locked exoplanets. Tidally locked planets are often slowly rotating, so their free troposphere could be in the WBG regime globally (Koll and Abbot 2016; Mills and Abbot 2013). These planets have one fixed diurnal hemisphere and one nocturnal hemisphere, corresponding to the moist and dry columns of our model, respectively. When the tidally locked planets are approaching the inner edge of the habitable zone, their surface temperature could be significantly higher than Earth’s tropical SST, providing an ideal environment for the vapor buoyancy feedback to work efficiently. However, previous studies have neglected the vapor buoyancy effect and assumed WTG (Yang et al. 2013; Yang and Abbot 2014; Pierrehumbert 2010a), which could lead to considerably narrower habitable zones. Therefore, we suggest that the vapor buoyancy effect should be accurately represented not only in GCMs but also in low-order models that are used to study climate habitability.

To focus on order-of-magnitude understanding, we have inevitably introduced simplifications to our model that only considers the clear-sky longwave radiation. An important one is that we use the two-band radiative transfer model, lacking detailed representation of water vapor’s absorption spectrum. We have also assumed that *β* is uniform in altitude, whereas *β* often has complicated vertical structures in the real atmosphere. However, a suite of cloud-resolving model (CRM) simulations has shown similar estimates of ΔOLR and *λ*. The CRM uses a comprehensive radiation scheme and explicitly simulates atmospheric circulation and water vapor dynamics. The CRM results have also shown that the vapor buoyancy effect does not affect the shortwave radiation budget and that the clear-sky effect dominates the OLR response. The CRM results, therefore, justify our simplifications and will be presented in a companion paper (Seidel and Yang 2020).

## Acknowledgments

This work was supported by Laboratory Directed Research and Development (LDRD) funding from the Lawrence Berkeley National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract DE-AC02-05CH11231, and used resources of the National Energy Research Scientific Computing Center (NERSC), also supported by the Office of Science of the U.S. Department of Energy, under Contract DE-AC02-05CH11231. This work was also supported by a Packard Fellowship for Science and Engineering awarded to author Yang. The authors thank Dr. Z. Tan and two anonymous reviewers for helpful comments and suggestions.

## APPENDIX

### Calculation Results Using a Real-Gas Radiative Transfer Model

We validate the simple model by using RRTM to compute ΔOLR and *λ*_{t}. We use the same temperature and water vapor profiles as in the simple model. To generate the model inputs, we interpolate these profiles onto 37 vertical levels from 0 to 1000 hPa. The simple model only considers a single greenhouse gas, water vapor. Therefore, we set all other trace gases, including CO_{2}, at zero concentration.

We use RRTM to separately compute OLR_{υ} and OLR_{nv}. We then estimate the radiative effect of vapor buoyancy as ΔOLR = OLR_{υ} − OLR_{nv}. From there we calculate *λ*_{t} according to Eq. (17). These RRTM results (Fig. A1) agree closely with the two-band calculation of the simple model in Fig. 4. This suggests that our estimates of ΔOLR and *λ* are robust.

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