Vapor buoyancy increases clear-sky thermal emission

The molar mass of H2O (18 g mol−1) is smaller than that of dry air (29 g mol−1), which makes humid air lighter than dry air with the same temperature and pressure. This vapor buoyancy (VB) effect has been traditionally considered small in large-scale climate dynamics and even neglected in some leading climate models. Here, using theory and aquaplanet simulations with prescribed surface temperatures, we show that VB increases tropospheric air temperature, and that the warmer atmosphere emits more clear-sky thermal radiation by about 2–4 W m−2 in the dry subtropical areas, a significant radiative effect. We then analyze Coupled Model Intercomparison Project Phase 6 simulations with prescribed sea surface temperatures and realistic topography. The results show that VB can increase clear-sky thermal radiation by up to 5 W m−2 over the ocean and about 15 W m−2 over the subtropical arid land regions. The radiative effect over arid land is amplified by a substantial increase of surface temperature due to VB. Our results highlight the role of VB in regulating Earth’s energy balance both at the top of the atmosphere and at the land surface. This study points to new ways to improve climate models and their simulated energy balance.


Introduction
Water vapor is one of the most potent greenhouse gases, maintaining Earth's surface temperature, and the latent heat release of water vapor condensation powers thunderstorms (Emanuel 1994, Pierrehumbert 2010. These two effects of water vapor are widely studied and acknowledged. However, water vapor has an overlooked effect on the density or buoyancy of an air parcel (Yang 2018a, 2018b, Betts and Bartlo 1991, Emanuel 1994. The molar mass of H 2 O (18 g mol −1 ) is smaller than that of dry air (29 g mol −1 ), which makes humid air lighter than dry air with the same temperature and pressure. To represent this effect, atmospheric scientists use virtual temperature to measure the density or buoyancy of an air parcel, which conveniently includes both temperature T and water vapor's specific humidity q. Since water vapor is considered a trace gas in the current climate, its mixing ratio is of a similar magnitude to specific humidity. Thus we do not distinguish between the two significantly in this paper. Virtual temperature is defined as where ν arises from the molar mass difference between water vapor and dry air, and its numerical value is about 0.61. Because the mixing ratio of water vapor is of O(10 g kg −1 ) in the most humid area on Earth, water vapor's buoyancy (VB) effect only causes a modest difference in atmospheric density. In other words, the virtual temperature is only a minor correction to temperature. As such, the impact of VB on atmospheric circulations and climate is not well-studied. However, it is the horizontal difference in density that controls the large-scale circulation, not its absolute magnitude. Recent studies showed VB contributes as much as temperature to the horizontal density differences (Nugent and Smith 2014, Yang et al 2022 according to theory, observations, and numerical models (figure 1). To facilitate the following discussion, we first define buoyancy using virtual temperature: Figure 1. (a) Schematic diagram for how VB makes sinking air warmer than rising air. (b) Reanalysis results averaged over June, July, and August: the black line represents total buoyancy in temperature unit (K), which is roughly horizontally uniform; the red line represents thermal buoyancy; the blue line represents VB.
where g is the gravitational acceleration, T V is the virtual temperature of an air parcel, and T V, R is the reference virtual temperature. If we take T V, R as the tropical averaged virtual temperature profile, B is close to 0 throughout the tropical free troposphere because the small Coriolis force cannot support strong gradients in density or virtual temperature (Charney 1963, Sobel et al 2002, Yang 2018b). Then we linearize equation (2) and get where ∆ h represents the horizontal difference between the air parcel and the reference profile. We call the first part thermal buoyancy, and the second part VB. That is, the sum of thermal buoyancy and VB remains constant in space. Since the tropical atmosphere is organized into moist convective regions (large VB) and dry subsiding regions (small VB), the dry regions must have substantial thermal buoyancy to compensate (figure 1(a)). Thus, the dry regions of Earth's tropics must be warmer than the moist regions. Reanalysis data supports this hypothesis (figure 1(b)). Figure 1(b) shows the mean meridional difference in buoyancy using temperature units at 700 hPa. The total buoyancy gradient is small, and the thermal buoyancy increase (red) is compensated by the reduction of VB (blue) away from the humid regions. This recent finding has broad implications for air temperature, thermal radiation, clouds, and climate change. Yang and Seidel (2020) hypothesized for the first time that VB makes the atmosphere of dry regions warmer than it otherwise would be, leading to enhanced clear-sky thermal emission. Using a 1D semi-analytic model, the authors showed that the radiative effect-the additional outgoing longwave radiation (OLR)-could be about 2-4 W m −2 in the current climate, and that this radiative effect can increase with climate warming. Seidel and Yang (2020) tested this hypothesis in a 2D cloud-resolving model using a regional-scale domain, with prescribed surface temperature. They found that clear-sky OLR was indeed greater in the simulation with VB than without. That difference in clear-sky OLR-the radiative effect of VB-also increases as the climate warms. This constitutes a negative climate feedback, which is distinct from other temperature feedbacks, such as Planck and lapse rate feedbacks. In the 2D CRM, Seidel and Yang calculated the feedback to be about O(0.15 W m −2 K −1 ) for the present tropical climate. This value is similar to the 1D estimate presented in Yang and Seidel (2020) and is comparable to net cloud feedbacks estimated using the state-of-the-art climate models (Zelinka et al 2020). Yang et al (2022) further showed in a 3D general circulation model (GCM) that VB makes the subtropical atmosphere warmer than it otherwise would be by about 1 K over a wide range of latitudes. This warming affects the lower tropospheric stability and thereby low cloud cover (e.g. the MD1 experiment in Yang et al (2022)). Unfortunately, some GCMs used to project future climates do not include VB in their atmospheric dynamical core (Yang et al 2022). Those GCM simulation results lack the moist-cold dry-warm temperature structure shown in figure 1. Yang et al (2022) showed that this error can reduce the modeled low cloud cover in the subtropics by about O(10%-30%), which is comparable to the low cloud bias between Coupled Model Intercomparison Project Phase 6 (CMIP6) models and observations (See figure S4 of Myers et al 2021). Therefore, VB's impact on low clouds is significant and may be critical to Earth's climate state and its climate sensitivity.
This study further explores the role of VB in 3D GCMs and quantifies its impact on the clear-sky OLR over the globe. This study complements earlier work that studied VB's radiative effect using 1D and 2D models, which did not include Earth's rotation, large-scale gradients in surface temperature and specific humidity, and the associated atmospheric circulations . This study will also quantify errors in the clear-sky OLR due to missing VB in GCM's dynamical core. Although this study uses similar data to Yang et al (2022), the objectives of the papers are distinct. Yang et al (2022) focused on VB's impact on clouds, but this paper focuses on VB's impact on clear-sky OLR. In section 2, we introduce the model, data, and analysis methods. In section 3, we perform a back-of-the-envelope calculation of VB's radiative effect. In sections 4 and 5, we show results from aquaplanet and CMIP6 simulation results. In section 6, we summarize and discuss our results.

Model
We perform the computer simulations using a 3D atmosphere model, known as AM2, from the Geophysical Fluid Dynamics Laboratory (GFDL) (Anderson et al 2004). The simulation setup is identical to that in Yang et al (2022). AM2 solves the governing equations of the atmosphere and uses a comprehensive set of physics packages, including a real-gas radiative transfer scheme, a modified Arakawa-Schubert convection parameterization, and a fully prognostic cloud scheme.
We perform a control (CNTL) simulation and a mechanism-denial (MD) simulation to study the effect of VB. Both simulations use a horizontal resolution of 2 • latitude × 2.5 • longitude and 25 sigma levels in the vertical. The lower boundary of AM2 is an ocean surface, whose temperature T s is prescribed as a function of latitude θ: This temperature distribution mimics Earth's climatological sea surface temperature (SST) distribution. We have removed the seasonal cycle and have fixed the solar insolation at its annual mean values. In MD, we have modified the dynamical core so that VB is not considered in the pressure gradient calculation, as in the NASA GISS model. Specifically, this is achieved by setting the molecular-weight ratio between water vapor and dry air as 1 in the model dynamical core, but not in physics schemes. The CNTL and MD simulations are otherwise identical. In the paper, we analyze data of the last five years of the 15 year simulations.

CMIP6 data
We analyze CMIP6 model output. Monthly outputs are obtained from historical simulations  of multiple models in the CMIP6 (Eyring et al 2016). The analyses focus on atmosphere-only simulations forced by the identical SST. We choose 23 different models, and the simulations follow the protocol of the Atmospheric Model Intercomparison Project (AMIP). The models include ACCESS-CM2, BCC-CSM2-MR, There are six models that appear to miss VB: NASA GISS-E2-1-G, CAS-ESM2-0, CNRM-CM6-1, FGOALS-g3, IITM-ESM, and IPSL-CM6A-LR.

Offline radiative transfer calculations
We calculate the clear-sky radiative effects due to perturbations in temperature and water vapor using two methods. First, for the AM2 simulations, we perform partial radiative perturbation (PRP) calculations (Colman and McAvaney 1997) with RRTMG (Mlawer et al 1997). That is, we perform offline radiation calculations in RRTMG for three thermodynamic profiles: (a) the CNTL water vapor and temperature, (b) the CNTL water vapor profile with the MD temperature profile, and (c) the MD water vapor profile with the CNTL temperature profile. Differencing the OLR given by (a) and (b), we obtain an estimate of the contribution to the radiative effect from the temperature difference between CNTL and MD. Differencing the OLR given by (a) and (c), we obtain an estimate of the contribution to the radiative effect from the moisture difference between CNTL and MD.
The second method is the radiative kernel method, which we use to calculate VB's radiative effect in CMIP6 models. We employed the clear-sky radiative kernels by Huang et al (2017). We decompose the clear-sky radiative effect into three components-due to perturbations in air temperature, water vapor, and surface temperature, respectively. Given perturbations in temperature or moisture, we can calculate its contribution to the clear-sky radiative effect by calculating the inner product of the perturbations and the radiative kernel. That is, we multiply the perturbations with the kernel and then perform a vertical integration.

Back-of-the-envelope calculation
We have shown that VB can increase air temperature by O(1 K). Here we show this temperature increase can lead to a significant OLR increase. Assuming the Black-Body radiation, then OLR is given by where σ is the Stefan-Boltzmann constant. When the VB-induced temperature increase ∆T is small, we can perform a first-order Taylor expansion and get Here ∆T (without subscript h) measures VB-induced change in T. If ∆T = 1 K and T = 255 K, VB-induced ∆OLR is about O(4 W m −2 ). In comparison, the radiative forcing due to 2 × CO 2 is about 4 W m −2 . Therefore, VB-induced ∆OLR or the radiative effect is significant. This calculation is highly idealized. We use blackbody radiation to represent atmospheric radiative transfer; we assume that VB only affects the temperature field, not atmospheric circulations nor specific humidity; we prescribe VB's warming as 1 K. However, this calculation provides an order-of-magnitude estimate as well as a testable hypothesis-VB has significant radiative effect. The following sections will show results from 3D GCMs. There, the above assumptions are relaxed, and we will test the hypothesis in a more realistic setting. Figure 2 shows the aquaplanet simulation results. Figure 2(a) shows the air temperature difference between the CNTL and MD simulations. We find that VB increases air temperature over a broad subtropical latitude range. The magnitude is larger in the lower free troposphere and can reach O(1 K). This agrees with the simple physical picture in figure 1(a): VB and thermal buoyancy compensate each other to maintain a uniform horizontal buoyancy distribution. In the lower troposphere, there is more water vapor, leading to greater horizontal moisture contrast, so the effect-temperature increase-is larger. Meanwhile, VB reduces specific humidity over the subtropics (figure 2(b)), making the atmosphere optically thinner. The reduction of specific humidity likely results from stronger subsidence. The warmer and drier atmosphere increases outgoing thermal radiation (figure 3) ∆OLR peaks at around 15 • away from the equator, and the magnitude is about 4 W m −2 . ∆OLR remains significant over a broad latitude range, suggesting that VB's radiative effect is global.

Aquaplanet simulations
Here, we further quantify contributions to ∆OLR using PRP calculations (figure 3). We find that VB-induced greater temperature increases OLR by about 1 W m −2 over the subtropics, with a more modest effect in the deep tropics (−5 • -5 • ) or higher latitudes (beyond 40 • off the equator). This magnitude agrees with our order-of-magnitude estimate, but it is significantly smaller than the total clear-sky ∆OLR in the subtropics. Changes in specific humidity are responsible for a substantial portion of ∆OLR and even dominate the overall meridional structure of ∆OLR. A drier atmosphere in the subtropics increases OLR by about 2 W m −2 , and the amplitude peaks at about 10 • latitude, where the total clear-sky ∆OLR reaches its maximum. In the global average, ∆OLR is 1.32 W m −2 , of which 0.82 W m −2 comes from a warmer atmosphere, 0.45 W m −2 comes from a drier atmosphere, and 0.05 W m −2 is a residual of the PRP calculation.
The aquaplanet simulation explicitly resolves large-scale atmospheric dynamics and employs a real-gas radiative transfer scheme. As such, the results may be more realistic and make important departures from our simple theory. For example, the current theory did not predict moisture change to induce a difference in OLR, which turns out to be significant in setting the amplitude and the spatial structure of the radiative effect.

AMIP simulations
We further test the hypothesis using the AMIP historical runs of the CMIP6 models. Yang et al (2022) analyzed 23 CMIP6 models and found that there are six models likely not representing VB in their dynamical core. For example, the NASA GISS ModelE does not include VB in its dynamical core. This is evident from the diagnostic results (figure 2 in Yang et al 2022) and the model description (Schmidt et al 2006). Here we analyze the same 23 models and divide them into Group A (with VB) and Group B (without VB). Then their contrast may be attributed to VB, analogous to comparing CNTL and MD in the aquaplanet simulations. Figure 4(a) shows the map view of the annually averaged temperature difference ∆T a between Groups A and B at 700 hPa. Similar to the aquaplanet simulations, VB significantly increases air temperature over a broad range of latitudes, not limited to the deep tropics. The temperature difference is relatively small along the Intertropical Convergence Zone (ITCZ) and gradually increases with latitude and reaches maximum about 2 K in the subtropical dry regions, consistent with our expectation. The temperature difference has a rich zonal structure due to the land-sea contrast and topography. The overall results are robust over summer and annual-mean climatologies (figures 4(a) and (b)). The results are also robust over different vertical levels, e.g. 500 hPa or 600 hPa. Figure 4(c) shows the map view of the difference in specific humidity ∆q between Groups A and B at 700 hPa. In the annual mean, the signal is primarily concentrated within −30 • to 30 • latitudes. VB decreases specific humidity over much of the Pacific and Atlantic basins and over the northern part of Africa, which could be related to enhance large-scale subsidence. VB also increases specific humidity along the coasts of South America, Africa, East Asia, and Australia. The magnitude of the specific humidity difference is less than 1 g kg −1 at 700 hPa. In northern hemisphere summer, the overall magnitude is stronger and reaches higher latitude in the northern hemisphere ( figure 4(d)). The overall results are robust over different vertical levels, e.g. 500 hPa or 600 hPa. Figure 4(e) shows the map view of the annually averaged difference in surface temperature ∆T s between Groups A and B. In AMIP runs, although SST is prescribed, surface temperature over land can evolve independently according to the surface energy balance of each GCM. VB increases land surface temperature by about 2-3 K over the southwestern US and 4-5 K over western China. In the northern hemisphere summer, this land surface temperature difference becomes even more prominent over the subtropics, accompanied by a cooler surface at higher northern latitudes and a warmer surface in Antarctica. The land surface warming was absent from our aquaplanet simulations with fixed SST and was also ignored in earlier studies of VB which used idealized boundary conditions. ∆T a , ∆q, and ∆T s can all contribute to changes in clear-sky ∆OLR (figure 5). Figures 5(a) and (b) shows the difference in clear-sky OLR between Groups A and B in the annual mean and in northern hemisphere summer, respectively. From −40 • to 40 • latitude, VB substantially increases clear-sky OLR with a strong land-sea contrast in ∆OLR: over oceans, ∆OLR can reach up to 4 W m −2 , while over land, ∆OLR can reach up to 18 W m −2 in the northern hemisphere summer. Given ∆T a is similar over ocean and land, this result indicates significant contribution from ∆T s and ∆q. ∆OLR is muted over higher latitudes in the annual mean but is still significant in summer.
Before we discuss further radiative transfer calculations, we perform a t-test to discern whether the clear-sky OLR is significantly different between Group A and Group B models. In figures 5(a) and (b), stippling indicates where the OLR difference due to VB is statistically significant with a p-value of <0.1. ∆OLR is statically significant over the tropical Eastern Pacific and tropical Atlantic. ∆OLR is also statically significant over subtropical arid land regions in North America, Northern Africa, and Asia. This result is consistent with our hypothesis that the radiative effect of VB is strongest in relatively dry regions.
Using radiative kernels, we decompose the total clear-sky OLR response into components involving ∆T a , ∆q, and ∆T s . We find that VB-induced warming increases clear-sky OLR over almost the entire planet (figures 5(c) and (d)). On average, the warmer atmosphere leads to about 2 W m −2 radiative effect from −40 • to 40 • latitude (figure 6). ∆q decreases OLR over much of the globe but increases OLR over limited areas in the tropical Atlantic and tropical eastern Pacific, where VB makes the atmosphere drier (e.g. figures 5(e) and (f)). Taking a zonal average, ∆q decreases OLR over all latitudes by about 1 W m −2 . This is a smaller contribution than that of ∆T a . ∆T s is more regionally confined than ∆T a and ∆q; thus, its radiative effect is regionally confined as well. Over the southwestern US and western China, ∆T s increases OLR by about 3 W m −2 and 7 W m −2 , respectively in the annual mean ( figure 5(g)). In summer, ∆T s makes even bigger contributions to ∆OLR (figures 5(h) and 6). With VB, the hotter surface emits more thermal radiation to space.
The kernel calculation shows an appreciable residual, though it remains substantially smaller than the ∆T a and ∆q contributions to ∆OLR. This residual may result from the fact that the Huang et al (2017) kernel was calculated from reanalysis data for 2008-2012. In contrast, the AMIP models achieve a range of climate states and cover a broader historical period. Their differences in temperature and water vapor of the base state may influence their sensitivity to changes in temperature and water vapor. The GCMs also use different radiative transfer schemes, which may also contribute to this residual.

Summary and discussions
Our results support the hypothesis that significant VB can make the atmosphere warmer than it otherwise would be, and that this higher temperature can increase OLR. We first used black-body radiation to perform an order of magnitude estimation: the radiative effect can be of O(4 W m −2 ) given a temperature increase of 1 K due to VB. We then estimate the VB-induced temperature increase and radiative effect using aquaplanet simulations. We perform a CNTL simulation and a MD simulation without VB. Their difference highlights the role of VB: VB can increase air temperature by 1-2 K in the lower troposphere in the subtropics. The warmer atmosphere can increase clear-sky OLR by about 1.5 W m −2 in the subtropics. Meanwhile, VB induces anomalous atmospheric circulations, making the subtropics drier. This effect further increases the clear-sky OLR by about 2.5 W m −2 , so the total clear-sky ∆OLR reaches up to 4 W m −2 in the subtropics.
Unfortunately, not all CMIP6 models incorporate VB accurately. This flaw leads to biases in simulated temperature and moisture, which further aggregate to cause bias in OLR or ∆OLR. Over the broad subtropical latitudes, the typical OLR bias reaches 2 W m −2 . For particularly arid regions, the OLR bias can reach about 10 W m −2 . Although different CMIP models may have different physics parameterizations, parameter values, and numerical schemes, leading to substantial OLR differences, our statistical analysis suggests that VB-induced OLR difference between Groups A and B is statistically significant over broad tropical ocean and land regions. Through offline radiative transfer calculations, we show that much of the clear-sky ∆OLR is due to VB-induced warming. Over subtropical arid regions, a warmer surface further increases OLR. On the other hand, greater specific humidity decreases OLR over much of the globe.
In a warmer climate, we expect to observe more water vapor in the atmosphere assuming a constant relative humidity. Thus VB-induced warming ∆T a would be greater than in the present climate, which can lead to a stronger radiative effect. This comprises a negative climate feedback that helps to stabilize Earth's climate. Our preliminary results show that this feedback is about 0.1 W m −2 K −1 in the current climate and increases exponentially with warming, reaching about 1 W m −2 K −1 at 310-320 K surface temperature . This feedback therefore attains leading-order importance in warm climates of the past (e.g. early Eocene) or, perhaps, the future. Therefore, missing VB will substantially impact GCM's capability in simulating climate change.
Although our statistical analysis of CMIP models is performed with a small sample, past work using a wide range of approaches confirms the robustness of the results. Together with Yang and Seidel (2020) and Seidel and Yang (2020), this paper examines VB's effect on Earth's thermal emission using a wide hierarchy of models. In this paper, we first calculated the radiative effect using the black-body radiation of a single-layer atmosphere (section 3). Yang and Seidel (2020) estimated the radiative effect using a two-band radiative transfer model coupled with a two-column atmosphere that is stably stratified. Then Seidel and Yang (2020) computed the radiative effect using a 2D, non-rotating cloud-resolving model with fixed SSTs and a real-gas radiative transfer scheme. This study further computed the radiative effect using 3D GCMs with real-gas radiative transfer schemes (sections 4 and 5). The overall results are robust: all calculations with different complexities show that VB increases air temperature and thereby clear-sky OLR.
In addition to providing more realistic results, gradually increasing the model complexity reveals important departures from simple models. These departures represent limitations of the simple models and raise new research questions awaiting to be addressed. For example, when it was first proposed, Yang and Seidel (2020) used an atmosphere with zero horizontal buoyancy gradient to illustrate the idea. That gave the impression that VB would only affect Earth's deep tropics, with ample moisture and small buoyancy gradient. However, we find that VB increases air temperature over a much wider latitude range in our aquaplanet simulations (figure 2(a)) and AMIP historical simulations (figures 4(a) and (b)). What determines the meridional extent of VB's impact in temperature, humidity, and OLR? VB affects the water vapor distribution in our GCM simulations, yet this effect was not considered in the simple, semi-analytic model . What controls the magnitude and sign of the water vapor response to VB? VB also increases land surface temperature, which influences surface evaporation, regional circulation patterns, and heat stress. What controls the magnitude and spatial structure of the land warming? Next steps should address the three basic questions and complement our current understanding of how VB affects Earth's climate.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https:// ucdavis.app.box.com/file/994275826488?s=crb3jch5h94tv4ahqzrywa2lal6dako7.