a. WBG dynamics

In this section, we ask whether the atmosphere is hotter due to VB in an environment with appreciable planetary rotation. That is, we are concerned with the difference in simulated temperature due to VB. We begin by obtaining a simple analytic prediction for that temperature difference under strict WBG conditions ΔT_wbg. Since density is approximately horizontally homogeneous in the tropical free troposphere under WBG (Charney 1963; Sobel et al. 2001), we equate the virtual temperature of a parcel of saturated air (T_υ,sat) and the virtual temperature of air in the simulated or observed atmosphere T_υ:

$T_{\upsilon }=T_{\upsilon ,\text{sat}}.$(3)

Substituting the definition of virtual temperature, we obtain

$T(1+\nu q)=T_{\text{sat}}[1+\nu q^{*}(T_{\text{sat}})].$(4)

We define ΔT_wbg = T − T_sat as the temperature difference due to VB. We further linearize $q^{*}$ around T using the Clausius–Clapeyron relation ${\partial }_T{q}^{*}=L{q}^{*}/R_{\upsilon }{T}^2$, where L is the latent heat of vaporization of water vapor and R_υ is the gas constant of water vapor. Substituting T_sat = T − ΔT_wbg into Eq. (5) and expanding around ΔT:

$T(1+\nu q)=T(1+\nu q^{*}(T))-\text{Δ}{T}_{\text{wbg}}\left[1+\nu q^{*}(T)+\nu \frac{L}{R_{\upsilon }T}{q}^{*}(T)\right]+\text{Δ}{T}_{\text{wbg}}^2\nu \frac{L}{R_{\upsilon }{T}^2}{q}^{*}(T).$(5)

We note that $\nu q^{*}(T)\ll 1$ and exclude the fourth term on the right-hand side. We also exclude the higher-order ΔT² term and then reorganize

$\text{Δ}{T}_{\text{wbg}}=\frac{\nu T[q^{*}(T)-q]}{1+\nu \left(\frac{L}{R_{\upsilon }T}\right)q^{*}(T)}.$(6)

We will now compare ΔT_wbg to the simulated differences in temperature between the CNTL and MD simulations:

$\text{Δ}{T}_{\text{MD-DYN}}=T_{\text{CNTL}}-T_{\text{MD-DYN}},$(7)

$\text{Δ}{T}_{\text{MD-ALL}}=T_{\text{CNTL}}-T_{\text{MD-ALL}},$(8)

where T_CNTL is the temperature of the atmosphere in the control simulations and T_MD-DYN and T_MD-ALL are the temperatures in the mechanism-denial simulations. Figures 2a and 2b show ΔT_MD-DYN and ΔT_MD-ALL, respectively, at the control surface temperature (300 K at the equator). The subtropical midtroposphere is up to 1 K warmer in the CNTL simulation than the MD-DYN simulation and up to 2.5 K warmer in the CNTL simulation than in the MD-ALL simulation.

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Temperature difference due to VB (color contours). (a) The ΔT_MD-DYN for an equatorial surface temperature of 300 K. (b) The ΔT_MD-ALL for an equatorial surface temperature for 300 K. (c) The ΔT_wbg for an equatorial surface temperature of 300 K. (d) The ΔT_MD-DYN for an equatorial surface temperature of 306 K. (e) The ΔT_MD-ALL for an equatorial surface temperature of 306 K. (f) The ΔT_wbg for an equatorial surface temperature of 306 K. The black contours represent the temperature of the CNTL simulations.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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Figures 2d and 2e show ΔT_MD-DYN and ΔT_MD-ALL, respectively, when the surface temperature is 6 K greater. Both ΔT_MD-DYN and ΔT_MD-ALL increase with climate warming.

Figures 2c and 2f show ΔT_wbg calculated from the control simulation. In the midtroposphere, ΔT_wbg is a close match to ΔT_MD-DYN (Figs. 2a,d), but substantially underestimates ΔT_MD-ALL. This is because ΔT_wbg captures the warming due to VB’s interaction with the large-scale dynamics, but it does not account for differences due to boundary layer turbulence or convection. For greater clarity, Fig. 3 compares ΔT_MD-DYN, ΔT_MD-ALL, and ΔT_wbg at the 691-hPa model level. The ΔT_wbg closely approximates ΔT_MD-DYN, particularly in the tropics (±30°). Even in a model with appreciable planetary rotation where strict WBG does not hold, VB nevertheless warms the atmosphere by the amount predicted by ΔT_wbg. This suggests that the vapor-buoyancy feedback is active in the real-world subtropics. We shall evaluate the simulated feedback later in section 4.

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Temperature difference due to VB at the 691-hPa model level, for the simulations with the equatorial surface temperature of 300 K.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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b. Moist adiabats and boundary layer dynamics

In this section, we shall address the discrepancy between ΔT_wbg and ΔT_MD-ALL. Figure 3 shows that ΔT_wbg captures the gross features and scale of the meridional pattern of ΔT_MD-ALL within the tropics: Both have local maximums around 15° and local minimums at the equator. However, ΔT_MD-ALL is considerably greater than either ΔT_wbg or ΔT_MD-DYN.

Additionally, ΔT_MD-ALL shows substantial warming in the upper troposphere, which is absent in ΔT_wbg and ΔT_MD-DYN. Free-troposphere temperature in the tropics is tightly linked to boundary layer moist static energy, as the virtual temperature profile is set by convection which ascends from there (Emanuel et al. 1994). CAM6’s Zhang–McFarlane convection scheme, like most deep convection schemes, is in fact formulated based on this relationship to boundary layer energy (Zhang and McFarlane 1995). Therefore, we will draw a link between VB’s influence on boundary layer dynamics and the additional warming indicated by ΔT_MD-ALL.

To examine the influence of the boundary layer, we calculate the temperatures of parcels undergoing moist ascent from there. Using an analytic expression provided by Romps (2017), we first calculate the lifting condensation level for an equator-average (±5°) parcel lifted dry-adiabatically from the 913-hPa model level. Then, we calculate the moist-adiabatic temperature profile T_ma for a parcel which ascends from that lifting condensation level. The horizontal axis of Fig. 4 shows the calculated moist-adiabatic temperature T_ma for each model level between 300 and 800 hPa in the same ±5° latitude band. The vertical axis is the actual simulated temperature T at that level. Filled circles denote the CNTL simulation, and open circles denote MD-ALL. Overall, T_ma is a strong predictor of T, as the marks run approximately parallel to the one-to-one line. More importantly, a difference in T_ma between CNTL and MD-ALL is matched by an approximately equal difference in T. This suggests that differences in boundary layer moist static energy are responsible for the large values of ΔT_MD-ALL at the equator. We may assume that those differences in free-troposphere temperature at the equator are then communicated to the subtropics via WBG dynamics, resulting in the large difference between ΔT_MD-DYN and ΔT_MD-ALL seen in Figs. 2 and 3.

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Equatorial region (±5°) free-troposphere temperature as predicted by moist-adiabatic ascent from the equatorial boundary layer (horizontal axis) plotted against the simulated difference in temperature. Filled circles denote data from the CNTL simulation, and open circles denote MD-ALL.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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How does VB cause the CNTL simulation to have a warmer boundary layer than the MD-ALL simulation? There are likely two factors at play. First, VB weakens the inversion strength within the boundary layer scheme, which allows greater entrainment of free-troposphere air to warm the equatorial boundary layer. Second, VB accounts for about 50% of surface buoyancy fluxes in Earth’s tropical ocean (Yang et al. 2022). The greater surface buoyancy fluxes in CNTL compared to MD-ALL increase turbulent kinetic energy production in the boundary layer (as shown in Fig. S2). This effect can then enhance the turbulent entrainment of free-troposphere warm air into the boundary layer. Our analysis will focus primarily on the first of these two effects, in which VB weakens the boundary layer inversion.

We begin by assessing the virtual potential temperature structure of the simulated lower troposphere as averaged around the equator (±5°). Figure 5a shows the virtual potential temperature θ_υ profile between 700 hPa and the surface for the three simulations with the equatorial surface temperature of 300 K:

${\theta }_{\upsilon }=T{\left(\frac{p_0}{p}\right)}^{R/c_p}(1+\nu q),$(9)

where p₀ = 1000 hPa is a reference pressure, p is atmospheric pressure, R is the gas constant for air, and c_p is the specific heat capacity of air at constant pressure. To imitate our treatment of the parameterizations, ν is set to zero for MD-ALL.

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Profiles of lower-troposphere, equator-region (±5°) (a) virtual potential temperature, (b) Δθ_υ,MD-ALL, (c) temperature, (d) relative humidity with respect to temperature, (e) specific humidity, and (f) moist static energy, for the simulations with equatorial surface temperature 300 K.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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Δθ_υ,MD-ALL is greater below 900 hPa (i.e., in the boundary layer) than it is above 800 hPa, by as much as 1 K. This indicates that the boundary layer inversion is weaker due to VB. We would expect a weaker boundary layer inversion to cause greater entrainment of free-troposphere air in CNTL than in MD-ALL. Due to vertical gradients in potential temperature and specific humidity, that entrainment should cause greater boundary layer temperature and reduced boundary layer relative humidity. Indeed, that is what we find in Figs. 5c and 5d: a hotter, less saturated boundary layer. However, the greater temperature in the CNTL boundary layer allows it to attain greater specific humidity q despite having lower relative humidity (Fig. 5e). As a result, the boundary layer moist static energy is greater in CNTL than in MD-ALL (Fig. 5f). The greater moist static energy in the CNTL simulation’s equatorial boundary layer is transmitted upward via convection, resulting in the greater free-troposphere temperature compared to MD-ALL, as seen in Fig. 4.

a. Clear-sky VB feedback

Here, we investigate the differences in simulated clear-sky outgoing longwave radiation ΔR_cslw between the CNTL and MD simulations. The green marks in Fig. 7a show tropical-average (±30° latitude) ΔR_cslw for MD-DYN. The value of ΔR_cslw increases as the climate warms, suggesting that the vapor-buoyancy feedback is active there. To attribute ΔR_cslw to VB, we must decompose this radiative effect according to its contributions from temperature and water vapor.

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Difference in net clear-sky OLR at top of atmosphere for (a) the MD-DYN experiment and (b) the MD-ALL experiment. Closed blue circles indicate the radiative effect implied by ΔT_υb as simulated by either experiment. Open blue circles indicate the radiative effect implied by ΔT_wbg. The red circles indicate the radiative effect implied by differences in specific humidity between the CNTL and MD simulations. The violet circles indicate the sum of the temperature and specific humidity effects. The green crosses indicate the simulated difference in clear-sky OLR. The blue curve indicates a cubic fit to the open circles.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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To separate ΔR_cslw into its contributions from tropospheric temperature and specific humidity, we use clear-sky approximate radiative kernels. These kernels are linear response functions of top-of-atmosphere radiation to perturbations in temperature and humidity. The kernels are described in appendix B. To obtain radiative effects, we take the inner products of the temperature and humidity kernels with the thermodynamic perturbations ΔT_MD-DYN and Δq_MD-ALL, using the lapse-rate tropopause² as the upper limit for the integral. The solid blue circles in Fig. 7a show the differences in ΔR_cslw due to tropospheric ΔT_MD-DYN, whereas the red circles show the differences due to tropospheric Δq_MD-DYN. The violet circles show the sum of these two radiative effect. These closely approximate ΔR_cslw, which suggests that stratospheric effects are negligible. Finally, the open blue circles indicate the temperature radiative effect using ΔT_wbg.

We can draw two inferences from Fig. 7a. First, VB is responsible for a robust negative feedback as shown by 1) the robustness of the trend in ΔR_cslw with warming and 2) that this trend can be explained well by ΔT_wbg. Second, there is essentially no countervailing change in the water vapor feedback due to VB: tropospheric water vapor does not contribute to the trend in ΔR_cslw. To supplement this result, appendix A discusses the differences in humidity between CNTL and MD-DYN.

Figure 7b is the same as Fig. 7a, but derived from the MD-ALL experiment. In this experiment, there is an additional positive water vapor feedback due to VB, indicated by the negative trend of the red marks with climate warming. However, this is offset by a greater negative feedback due to ΔT_MD-ALL. The sum of the temperature and water vapor radiative effects in the MD-ALL experiment indicates a negative VB feedback similar both to MD-DYN and to the vapor-buoyancy feedback implied by ΔT_wbg. Notably, the MD-ALL experiment shows a larger radiative effect of ΔT_MD-ALL (solid blue circles) than is predicted by WBG theory (open blue circles). This can be attributed to the additional difference in atmospheric temperature for MD-ALL discussed in the previous section. The warmer atmosphere emits more radiation to space. The greater warming is partially offset by additional atmospheric water vapor, as evidenced by the comparatively smaller discrepancy between WBG theory (open blue circles) and the combined radiative effect of ΔT_MD-ALL and Δq (solid violet circles).

The MD-DYN and MD-ALL experiments exhibit similar differences in clear-sky feedback compared to CNTL. The total difference in clear-sky feedback in MD-ALL, Δλ_cslw,MD-ALL, can be estimated by linear regression as 0.20 W m⁻² K⁻¹, compared to 0.22 W m⁻² K⁻¹ for Δλ_cslw,MD-DYN. Evidently, any additional trend in temperature due to the boundary layer mechanisms discussed in section 3b is offset by additional atmospheric water vapor. This is because greater temperature at the sites of deep convection causes greater water vapor detrainment (due to Clausius–Clapeyron). That additional water vapor has a strong countervailing effect on OLR (Jeevanjee et al. 2021).

We refer to the climate feedback derived from the trend in ΔT_wbg as the VB feedback. The magnitude of the VB feedback is measured by its feedback parameter, which represents a top-of-atmosphere flux sensitivity to a unit increase of surface temperature:

${\text{λ}}_{\upsilon b}=\frac{d{R}_{\upsilon b}}{d{T}_s},$(15)

where R_υb is the top-of-atmosphere radiative effect (additional OLR) due to ΔT_wbg. As before, we define upward fluxes as positive; therefore, a positive value of λ_υb indicates a negative climate feedback. To calculate λ_υb, we fit a cubic curve to the ΔT_wbg marks in Fig. 7 and take its derivative. We chose a cubic model to capture the nonlinear increase in feedback magnitude found in SY20. Figure 8 shows λ_υb plotted in red. At an Earth-like equatorial temperature of 300 K, the tropical-average feedback due to VB is 0.11 W m⁻² K⁻¹. This is about 5% of the total clear-sky longwave feedback of 2.2 W m⁻² K⁻¹ in the CNTL simulation. For context, both idealized column models and comprehensive GCMs similarly find a typical clear-sky longwave feedback of about 2 W m⁻² K⁻¹ (Zhang et al. 2020; McKim et al. 2021). The VB feedback increases strongly as the climate warms, reaching a value of 0.36 W m⁻² K⁻¹ at the greatest surface temperature we tested. Therefore, the VB feedback may play an outsize role in hot paleoclimates and hot planetary climates.

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Tropical-average VB feedback derived from a hierarchy of climate models. The CRM result is reported in Seidel and Yang (2020), and the simple column result is reported in Yang and Seidel (2020).

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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Figure 8 also plots data from two other studies of the VB feedback. The blue curve represents the feedback parameter as estimated by a curve fit in the SY20 study, which employed essentially the same feedback analysis methods as this study, albeit for an idealized 2D cloud-resolving model. The violet curve shows an estimate from a simple 1D column model described in Yang and Seidel (2020). That model employed a simple two-band radiation calculation for an all-troposphere idealized column with and without VB-induced warming. Here, we employ the calculations from that study with the that model’s humidity parameter set to β = 0.5. These curves represent a hierarchy of models for the VB feedback: a 3D GCM, a 2D CRM, and 1D column model. The three models broadly agree upon the magnitude of the VB feedback and its trend with warming, thus corroborating one another. The differences among these three models are due to differences in virtual temperature profiles, in their humidity profiles, and in their treatments of atmospheric radiation.

b. All-sky and global-average feedbacks due to VB

Our analysis of feedbacks due to VB has so far been limited to clear-sky radiation in the tropics. However, it is useful to ask whether VB alters the total climate feedback in a more complete picture: in the global average, with cloud radiative effects included. Figures 9a and 9b show the globally averaged differences in top-of-atmosphere all-sky radiation between CNTL and MD-DYN and CNTL and MD-ALL, respectively. As in the previous sections, we adopt a sign convention that upwelling fluxes are positive so that a positive trend indicates a negative climate feedback. For MD-DYN compared to CNTL, the all-sky radiative effect is greater than the clear-sky radiative effect by 4–6 W m⁻². The all-sky radiative effects also exhibit a greater trend (i.e., feedback) with warming. We can estimate the feedback magnitude as 0.41 W m⁻² K⁻¹ by way of a linear regression of radiative effect against surface temperature. This is greater than the clear-sky longwave difference of 0.17 W m⁻² K⁻¹. Calculating feedbacks in the same way for MD-ALL, the total all-sky feedback is 0.31 W m⁻² K⁻¹, compared to 0.14 W m⁻² K⁻¹ for clear-sky longwave. In either case, cloud feedbacks add to the vapor buoyancy feedback.

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Global-mean difference in net all-sky radiation (red marks) at top of atmosphere for (a) the MD-DYN experiment and (b) the MD-ALL experiment. Global-mean difference in clear-sky OLR (blue marks) are shown for comparison.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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The solid lines in Fig. 10 show Δλ_lwcld,MD-DYN and Δλ_swcld,MD-DYN. These shortwave and longwave feedbacks approximately cancel one another near the equator. That suggests a high-cloud effect, as greater cloudiness in the upper troposphere has similar but opposite effects on longwave emission and shortwave reflection. However, in the subtropics, a negative cloud shortwave feedback dominates the longwave. That may indicate a difference in low-cloud feedback between CNTL and MD-DYN. Subtropical low clouds are likely to become less extensive as the climate warms (Sherwood et al. 2020; Myers et al. 2021), and our aquaplanet simulations replicate that finding (not shown). The present results suggest that VB causes low clouds to recede less than they otherwise would, causing a more-negative climate feedback. This may be a consequence of the greater midtroposphere temperatures due to VB (Fig. 2), which cause VB to enhance the inversion strength as “seen” by the boundary layer scheme. That would promote subtropical boundary layer clouds by reducing the entrainment of dry free-troposphere air (Yang et al. 2022). If this generalizes to other models and configurations, those comprehensive GCMs which neglect VB in their pressure gradient may achieve a somewhat more negative cloud feedback if they were to instead incorporate VB into their dynamics.

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Simulated differences in cloud feedbacks due to VB.

Citation: Journal of Climate 37, 22; 10.1175/JCLI-D-23-0558.1

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The dashed lines in Fig. 10 show Δλ_lwcld,MD-ALL and Δλ_swcld,MD-ALL. Here, the deep tropical difference in cloud shortwave feedback is greater in magnitude than its counterpart longwave feedback. Since upper-troposphere clouds dominate there, this difference in magnitude may be due to differences in cloud altitude or in cloud optical depth. A more extensive feedback analysis would be necessary to distinguish these mechanisms. The differences in subtropical cloud feedbacks in MD-ALL are qualitatively similar to what we found for the MD-DYN experiment: VB induces a negative shortwave feedback without much change in longwave. As before, low clouds are presumably responsible for this behavior. However, the mechanism at work is likely different from MD-DYN. One possibility is that VB increases cloud development within the boundary layer by increasing the buoyancy of shallow convection (Dagan et al. 2018).