Quantifying the deviation of the tropical upper tropospheric temperature response to surface warming from a moist adiabat

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Introduction
One of the earliest general circulation model (GCM) predictions of the response to increased CO 2 is amplified warming aloft in the tropics (Manabe & Wetherald, 1975;Manabe & Stouffer, 1980).This prediction has since been confirmed by observations (Santer et al., 1996;Thorne et al., 2011;Flannaghan et al., 2014) and state-of-the-art models such as coupled Atmosphere-Ocean GCMs (AOGCMs) (Vallis et al., 2015) and cloud resolving models (CRMs) (Lau et al., 1993;Romps, 2011).The tropical temperature response has important implications for the global climate, as it sets the 1) static stability in the tropics, which influences the strength of deep convection (Singh & O'Gorman, 2013;Seeley & Romps, 2015), 2) meridional temperature gradient, which influences the position of the Hadley Cell edge and subtropical jet (Shaw et al., 2016), and 3) lapse rate feedback in the tropics, which exerts a strong influence on the global climate sensitivity owing to the large contribution of the tropics to the global mean (Popke et al., 2013;Po-Chedley et al., 2018).Amplified tropical upper tropospheric warming in response to increased CO 2 is predicted from the adjustment of a moist adiabat (Held, 1993).In particular, for a 4 K warming at the surface with fixed relative humidity, moist adiabatic adjustment predicts warming aloft of 10 K.While the moist adiabatic prediction is intuitive, it does not consider many other processes that may influence the temperature response to warming, such as the large-scale circulation, the direct effect of CO 2 , and convective entrainment.Emanuel et al. (1994) show that in the presence of a strong large-scale circulation, the free troposphere and the sub-cloud layer become decoupled in regions of climatological descent.Thus, we expect moist adiabatic adjustment to apply only over regions of deep convection.Brown and Bretherton (1997), Flannaghan et al. (2014), andFueglistaler et al. (2015) use precipitation-weighting to show that observed temperature trends in the upper troposphere are more strongly linked to surface trends in regions of deep convection.Andrews and Webb (2018) further demonstrate the importance of the large-scale circulation on the tropical warming response by showing in the HadGEM2-A model that localized SST warming in the western Pacific (where there is climatological deep convection) results in a warming response with strong amplification aloft, whereas the SST warming in the eastern Pacific (climatological descent) leads to warming confined below the tropical inversion.As Andrews and Webb (2018) focus on the role of the tropical temperature response on the lower-tropospheric stability, they do not quantify the deviation of the temperature response from a moist adiabat.
We do not expect the direct effect of CO 2 to lead to moist adiabatic adjustment because it does not impact the global-mean surface temperature.However, it does impact the large-scale circulation and tropical precipitation response (Bony et al., 2013;Merlis, 2015), and the tropospheric warming due to the direct effect is nearly uniform in height (He & Soden, 2015;Wang & Huang, 2020).Thus, we expect the moist adiabat to overpredict the temperature response in the presence of the direct effect of CO 2 .
We expect convective entrainment to weaken the amplification of warming aloft compared to a moist adiabat, as an entraining parcel releases less latent heat.Singh and O'Gorman (2013) and Seeley and Romps (2015) show that the increase in convective available potential energy (CAPE) with warming as obtained from CRMs is consistent with that predicted by the zero-buoyancy bulk-plume model.The zero-buoyancy bulk-plume model is a simple model for the tropical temperature profile that includes the effect of climatological convective entrainment.As CAPE quantifies the deviation of a temperature profile from a moist adiabat, increasing CAPE with warming is consistent with the overprediction of upper tropospheric warming by the moist adiabatic adjustment theory.Although previous studies have implied convective entrainment as an explanation for the overprediction of upper tropospheric warming by the moist adiabat (Tripati et al., 2014;Po-Chedley et al., 2019), the influence of varying entrainment rates on the temperature response in GCMs has not yet been reported in the literature.
Here, we quantify the moist adiabatic prediction in response to warming across the CMIP5 model hierarchy.We show that the moist adiabat overpredicts the modeled temperature response.We quantify the importance of three mechanisms on the overprediction of the moist adiabat: 1) the large-scale circulation, 2) the direct effect of CO 2 , and 3) convective entrainment.We quantify the importance of convective entrainment by varying the parameterized entrainment rate in idealized aquaplanet simulations.

CMIP5 models
We examine the tropical temperature response to warming across the climate model hierarchy using CMIP5 data (Taylor et al., 2012).At the most complex end, we consider the AOGCM response to a quadrupling of CO 2 (abrupt4×CO 2 ) relative to a pre-industrial climate (piControl) in 29 models (Supplementary Table S1).We average the last 30 years of the 150-year simulation to study the near-equilibrium response.
In the mid-range of complexity, we consider 11 atmospheric GCMs (AGCMs, see Supplementary Table S1) that prescribe the sea-surface temperature (SST) according to observations from 1979 to 2008 following the AMIP protocol (Gates, 1992).The indirect effect of CO 2 increase is quantified by imposing: 1) spatially-varying SST warming based on the CMIP3 multi-model mean response (amipF) and 2) uniform SST warming of 4 K (amip4K).This allows us to study the importance of patterned SST warming.We also consider the direct effect of increased CO 2 in the absence of SST changes (amip4×CO 2 ) and we add the direct and indirect effects to get the total response to increased CO 2 (amipF/4K+4×CO 2 ) that can be compared to the AOGCM response.We take the average over the entire 30 years of each simulation.
Finally, at the simple end we consider 9 aquaplanet AGCMs (see Supplementary Table S1).The indirect effect is quantified as a response to a uniform SST warming of 4 K (aqua4K) relative to the aquaplanet configured with the qObs SST profile (aqua-Control) (Neale & Hoskins, 2000).We also consider the direct effect of increased CO 2 (aqua4×CO 2 ) and add it to the indirect effect to get the total response to increased CO 2 in the aquaplanet (aqua4K+4×CO 2 ).

GFDL AM2.1 aquaplanet GCM
In order to understand the importance of entrainment for the tropical temperature response to surface warming we configure the GFDL AM2.1 aquaplanet GCM (hereafter GFDL) with the Relaxed Arakawa-Schubert (RAS) convection scheme (Moorthi & Suarez, 1992).In the RAS scheme, the Tokioka parameter (α) controls the minimum entrainment rate ( min ) as follows: where D is the depth of the planetary boundary layer.This constraint only affects plumes that detrain above 500 hPa, thus the Tokioka parameter controls the entrainment rate of deep convection only.Tokioka et al. (1988) varied α to study the influence of convective entrainment on the Madden-Julian oscillation.The default climatological value is α = 0.025 in GFDL.To investigate the role of entrainment on the tropical temperature response, we perturb α from its default climatological value as follows: α = 0, 0.00625, 0.0125, 0.05, and 0.1.
As varying α only indirectly affects the actual entrainment rate in the model, we quantify the entrainment rate using the output from the RAS scheme.The bulk entrainment rate is then calculated as the entrainment rate vertically averaged from 850-200 hPa.The expectation is that as convective entrainment rate increases (increasing α), the convecting plume becomes more sub-saturated, latent heating decreases, and the temperature response to surface warming weakens in the upper troposphere.
We vary the entrainment in two configurations of the GFDL model: 1) the standard aquaplanet configured with the qObs SST profile (GFDLaqua) (Neale & Hoskins, 2000) and 2) rotating radiative-convective equilibrium (RCE) configured with a spatially uniform SST of 300 K (GFDLrce).The latter allows us to test for the robustness of our results in the absence of a large-scale circulation, which is a common idealized model configuration for the tropics (Wing et al., 2018).For both configurations we investigate the response to a uniform SST warming of 4 K (GFDLaqua4K and GFDLrce4K).Following Tan et al. (2019) the GFDL aquaplanet uses RRTMG radiation and does not include the radiative effects of ozone and clouds.
We compare the tropical temperature response to warming with varying climatological entrainment in the aquaplanet to the zero-buoyancy bulk-plume models of Singh and O'Gorman (2013), hereafter SO13, Romps (2014), hereafter R14, and Romps ( 2016), hereafter R16.The zero-buoyancy bulk-plume model is a simple 1-D model that includes the effect of convective entrainment in RCE.The SO13 model assumes a fixed environ-mental relative humidity, while the R14 and R16 models explicitly consider the water vapor budget to predict relative humidity, which is further assumed to be vertically constant in R16.For the SO13 model, we assume a constant relative humidity profile of 80%.
For the R14 model, we assume a constant ratio of gross evaporation to gross condensation of 0.75 (α in R14) as this gives a close fit to both the GFDLrce and SO13 results.
For the R16 model, we assume a constant precipitation efficiency of 0.25 (PE in R16) to be consistent with the value of gross evaporation to condensation rate chosen for the R14 model.We configure all other parameters using the same values as reported in the literature.

Calculating the moist adiabat and its overprediction
We calculate the moist adiabatic temperature by setting the initial condition of the rising parcel as the annual mean 2 m temperature, humidity, and surface pressure.For models where the 2 m fields are not available, we interpolate the three dimensional temperature and humidity fields to the surface pressure.Where the surface pressure is greater than the lowest pressure level of the vertical grid (1000 hPa), we linearly extrapolate from the 1000 hPa value.
We integrate the dry adiabatic lapse rate Γ d up to the lifted condensation level (LCL).
During this dry ascent, we assume that the water vapor mixing ratio is conserved.Above the LCL, we calculate temperature by integrating the moist-adiabatic lapse rate Γ m following the definition in the American Meteorological Society (AMS) glossary (AMS, cited 2020: Moist-adiabatic lapse rate).
where L v is the latent heat of vaporization, r v is the vapor mixing ratio, R is the specific gas constant of dry air, R v is the specific gas constant of water vapor, T is temperature, and c pd is the isobaric specific heat capacity of dry air.This moist adiabat is a simplified form of a moist pseudoadiabat where it is assumed that all condensates precipitate out immediately and r v 1.Furthermore, we do not consider the effect of freezing (latent heat of fusion).
We quantify the overprediction O p of the moist adiabatic response at a pressure level p as follows: where ∆ denotes the difference between the warmer and climatological climates, T p is the GCM temperature at pressure level p, T m,p is the moist adiabatic temperature at pressure level p, and T s is the surface temperature.We evaluate overprediction at 300 hPa following Fueglistaler et al. (2015).The tropical-mean overprediction is obtained from horizontally-averaging between 10 • S and 10 • N.
To test the impact of the large-scale circulation, we average overprediction only over regions of climatological ascent at 500 hPa that exceeds the 75th percentile value in the tropics following Sherwood et al. (2014).This corresponds to ≈ −35 hPa/d in the multimodel mean climatology of the piControl and AMIP simulations.The overprediction in regions of deep convection is then obtained from the horizontally-averaged overprediction within regions that satisfy the 75th percentile pressure velocity criteria.We use −35 hPa/d as the threshold value across all models.We do not filter the GFDLrce response by vertical motion due to the absence of a large-scale circulation.

Overprediction across the CMIP5 model hierarchy
Moist adiabatic warming systematically overpredicts the multi-model mean upper tropospheric warming across the CMIP5 model hierarchy (red bars in boxes Fig. 1a).According to a t-test, the difference in mean overprediction between abrupt4×CO 2 and the simpler models is statistically significant at the 5% level (Supplementary Table S2).The multi-model mean overprediction varies by a factor of 2 across the model hierarchy, from 25.3% for abrupt4×CO 2 to 16.6%, 17.0%, and 12.9% for amipF, amip4K, and aqua4K, respectively.The overprediction is largest in the upper troposphere (Supplementary Fig. S1) and is similar for alternative definitions of moist adiabats, such as the pseudoadiabat and the reversible adiabat (Supplementary Table S3).
In what follows we focus on quantifying the impact of the following mechanisms on overprediction: 1) large-scale circulation, 2) direct effect of CO 2 , and 3) convective entrainment.

Large-scale circulation
The moist adiabatic prediction does not take into account the presence of the largescale climatological circulation or its response to warming.Since the moist adiabat is a model of a convecting parcel, we expect overprediction to be smallest over regions of deep convection (defined here as regions where climatological ω < −35 hPa/d at 500 hPa).
Overprediction is small in regions of deep convection such as the western Pacific warm pool (Fig. 2a-d When averaged only over regions of deep convection, multi-model mean overprediction decreases to 19.3%, 9.3%, and 13.4% for abrupt4×CO 2 , amipF, and amip4K (Fig. 1b).
This decrease is statistically significant at the 5% level (Supplementary Table S4).In contrast, the multi-model mean overprediction over regions of deep convection for aqua4K slightly increases to 13.1%, but this increase is not statistically significant.Clearly, the climatological large-scale circulation has an influence on the tropical temperature response, but accounting for this does not eliminate overprediction.

Direct effect of CO 2
The direct effect of increased CO 2 has a significant impact on the tropical circulation and precipitation but does not lead to significant global-mean surface warming (Bony et al., 2013).When the response to the direct effect of CO 2 is added to the surface warming effect in the AGCM and aquaplanet models, the multi-model mean overprediction over regions of deep convection increases to 20.1%, 21.1%, and 16.6% for amipF, amip4K, and aqua4K, respectively (compare Fig. 1b to Fig. 3), and the AMIP model results become more similar to CMIP5 models.A t-test shows that this increase is statistically significant at the 5% level for all three model configurations (Supplementary Table S5).Thus, the direct effect of CO 2 contributes to a non-zero overprediction as expected from previous work that showed the tropical temperature response to the direct effect of CO 2 is vertically uniform (compare vertical structure of black and orange lines in Supplementary Fig. S2).

Convective entrainment
Even after accounting for the large-scale circulation and the direct effect of CO 2 , overprediction is still non-zero (as shown by the AMIP and aquaplanet model results in Fig. 1b).This motivates us to consider the role of entrainment on overprediction, another mechanism that is missing in the moist adiabatic prediction.We study how the strength of climatological entrainment in the RAS convection scheme affects the magnitude of the overprediction in the GFDL model.With the default Tokioka parameter (α = 0.025), the moist adiabat overpredicts the GFDLrce4K and GFDLaqua4K response by 11.6% (Supplementary Table S3) and 13.2% (Supplementary Table S6), respectively.
The magnitude of overprediction in GFDL is similar to that of the CMIP5 aqua4K multimodel mean, making GFDL a good representative model for this study.
When the Tokioka parameter is increased and thus there is a larger entrainment rate, the temperature response is weakened aloft in both the RCE (Fig. 4a) and aquaplanet (Fig. 4b) configurations.The range of the overprediction obtained from varying the climatological entrainment rate in GFDLrce4K (GFDLaqua4K) is 6.7% to 17.1% (8.3% to 17.9%).Increasing α beyond the range shown here does not further increase the entrainment rate.Thus, the range of bulk entrainment rates obtained here represent nearly the full extent of the entrainment rate regime that can be studied by perturbing the Tokioka parameter in GFDL.
We find that overprediction is strongly correlated with the logarithm of the climatological entrainment rate for both GFDLrce4K (R = 0.95, see Fig. 4c) and GFDLaqua4K (R = 0.98, see Fig. 4d).While the range of overprediction obtained in GFDLaqua4K is similar to that of GFDLrce4K, GFDLaqua4K exhibits larger entrainment rates given the same Tokioka parameter.The sensitivity of overprediction to the strength of climatological entrainment obtained in GFDLrce4K is consistent with the zero-buoyancy bulkplume models of SO13 and R14 up to = 0.1 km −1 (dashed and solid black lines in

Summary
Here, we investigate the accuracy of the moist adiabatic prediction of the tropical upper tropospheric temperature response to warming.We found that the moist adiabat overpredicts the multi-model mean tropical upper tropospheric warming at 300 hPa by 12.9-25.3%across the CMIP5 model hierarchy.We quantified the importance of three mechanisms, not included in the moist adiabat theory, to the overprediction: 1) largescale circulation, 2) direct effect of CO 2 , and 3) convective entrainment.The importance of convective entrainment was quantified by varying the Tokioka parameter in idealized aquaplanet simulations.Our conclusions are: 1.The climatological large-scale circulation has a significant impact on overprediction.Overprediction is largest in regions of descent and weak ascent.Overprediction is smaller but non-zero in tropical regions of deep convection.This explains why multi-model mean overprediction is higher for the amip4K response (17.0%) compared to the aqua4K response (12.9%), which does not include climatological descent in the deep tropics (10 • N/S).
2. The direct effect of increased CO 2 , which impacts tropical circulation and precipitation but not global-mean warming, contributes significantly to overprediction.
This explains why multi-model mean overprediction is higher for the abrupt4×CO 2 response (25.3%) compared to the configurations with prescribed surface warming (16.6% for amipF, 17.0% for amip4K, and 12.9% for the aqua4K).rate in the RCE configuration agrees well with the zero-buoyancy bulk-plume models of Singh and O'Gorman (2013) and Romps (2014).The Romps (2016) model does not agree as closely.This may be due the additional simplifying assumptions that it makes about the vertical structure of entrainment, detrainment, and relative humidity.

Discussion
While Tripati et al. (2014) and Po-Chedley et al. (2019) attribute the overprediction of the moist adiabat to convective entrainment, our results show that the large-scale circulation and the direct effect of CO 2 also contribute to overprediction.This suggests that the predictions made by the zero-buoyancy bulk-plume models may have limitations outside of the idealized RCE configuration.Indeed, while the sensitivity of overprediction to climatological entrainment in the GFDL RCE aquaplanet agrees well with the zero-buoyancy bulk-plume model, this is not the case for the GFDL aquaplanet with a large-scale circulation.Future work could evaluate the bulk-plume model of Singh et al. (2019), which improves on the Singh and O'Gorman (2013) and Romps (2014) models by also considering the effect of the large-scale vertical motion on the predicted temperature response.
In our study, we perturbed the entrainment rate in an aquaplanet model by an order of magnitude but were not able to capture the full intermodel spread among the aqua4K models.Some possible reasons that our perturbation experiment failed to capture the full spread of overprediction include: 1) the RAS convection scheme is not used by all CMIP5 aquaplanet models and other convection schemes may show greater sensitivity to entrainment, 2) the entrainment response to warming (rather than the climatological entrainment) may influence overprediction, and 3) physical processes other than entrainment may influence overprediction.The importance of 1) may be addressed by running experiments using a different convection scheme that more explicitly allows the entrainment rate to be controlled.The importance of 2) may be quantified by prescribing different entrainment rates in a warmer climate.Prescribing different Tokioka parameters in the control and warm climates of the GFDL aquaplanet leads to a large range of overprediction (−40.4%-73.5%,see Supplementary Fig. S3).However, parameterized entrainment must be compared to more direct measures of entrainment such as those diagnosed from cloud-permitting model simulations (Romps, 2010).Future work could also explore 3) by quantifying the influence of other processes that are not represented in a moist adiabat on overprediction, such as precipitation efficiency, the ice phase, and cloud radiative effects.
This work highlights the limitations of moist adiabatic adjustment as a quantitative theory for the tropical temperature response predicted by climate models, and provides a first step towards a mechanistic understanding of this misfit.A full understanding of tropical lapse rate changes is critical to determine the robustness of model predictions, and to provide confidence in tropical climate forecasts more generally.
, inside the red contour line).Conversely, overprediction is large over the eastern Pacific, which is characterized by climatological descent (Fig.2a-d, regions outside of red contour line).Overprediction over the eastern Pacific is smaller in amip4K compared to amipFuture, suggesting that enhanced future warming in the eastern Pacific contributes to overprediction.Overprediction is zonally uniform in aqua4K (Fig.2e) and nearly meridionally uniform as most of 10 • N/S is a region of climatological deep convection in the aquaplanet.

Figure 1 .
Figure 1.a) Intermodel spread of overprediction across the CMIP5 model hierarchy.For each model configuration, black dots denote overprediction of individual models, the red horizontal line is the mean, the red vertical bar is the 5-95% confidence interval of the mean, and the blue vertical line is the standard deviation.b) Same as a), but overprediction averaged only over regions of deep convection (defined as where ω < −35 hPa/d at 500 hPa).

Figure 2 .
Figure 2. a) Spatial structure of the overprediction of the moist adiabat at 300 hPa in response to warming for the CMIP5 multi-model mean.The red contour denotes the boundary of the multi-model mean climatological deep convection as described in the text.b)-e) are the same for the amipF+4×CO2, amipF, amip4K, and aqua4K multi-model mean responses, respectively.

Figure 3 .
Figure 3. Same as Fig. 1b but including the direct effect of CO2 in the AMIP and aquaplanet model results.

Fig
Fig.4c).The R16 model predicts weaker overprediction for a given climatological entrainment rate compared to GFDLrce, SO13, and R14.The R16 prediction does not change substantially with varying values of precipitation efficiency.

3.
Parameterized convective entrainment contributes significantly to overprediction in the GFDL aquaplanet model configured with various Tokioka parameters.Overprediction scales with the logarithm of the climatological entrainment rate in the GFDL model.The sensitivity of overprediction to the climatological entrainment

Figure 4 .
Figure 4. Temperature response in the GFDL aquaplanet when varying the Tokioka parameter for the a) RCE (GFDLrce4K) and b) aquaplanet (GFDLaqua4K) configurations.Overprediction of the moist adiabat increases with the strength of climatological entrainment for c) GFDLrce4K and d) GFDLaqua4K.The deviation as predicted by zero-buoyancy bulk-plume models of Singh and O'Gorman (2013) (labeled SO13), Romps (2014) (labeled R14), and Romps (2016) (labeled R16) are shown as black lines in panel c.

Figure
Figure S2.a) Vertical structure of the difference in multi-model mean temperature response between amipF+4×CO2 and amipF (black) and the corresponding moist adiabatic prediction (orange).While the warming due to the direct effect of CO2 is approximately uniform with height in the multi-model mean, the moist adiabat predicts amplified warming aloft.b) and c) are the same for the differences between amip4K+4×CO2 and amip4K and aqua4K+4×CO2 and aqua4K, respectively.

Figure S4 .
Figure S4.The difference between overprediction averaged over 10 • N/S and overprediction averaged only over regions of climatological deep convection (ω500 < −35 hPa/d) for each model across the model hierarchy (black dots).The mean difference in overprediction is denoted by the red line.The red box shows the 5-95% confidence interval of the mean.The blue line shows one standard deviation of the distribution.

Figure S5 .
Figure S5.The difference in overprediction between the combined surface warming plus the direct CO2 response and only the surface warming response for each model across the model hierarchy (black dots).The mean difference in overprediction is denoted by the red line.The red box shows the 5-95% confidence interval of the mean.The blue line shows one standard deviation of the distribution.

Table S2 .
P-values of the T-test for the null hypothesis that the difference in mean overprediction between the abrupt4×CO2 response and that of simpler models are indistinguishable.The mean difference and the 5-95% confidence interval are also shown.The difference is statistically significant for all model configurations (p-value < 5%, indicated in bold).

Table S3 .
Overprediction in % of the moist adiabat across the model hierarchy for various types of the moist adiabat.Three types of moist adiabats are shown here following the definitions in the AMS glossary.Standard : The limit of a moist pseudoadiabat when rv 1 (AMS, cited 2020: Moist-adiabatic lapse rate).Pseudo: Moist pseudoadiabat, which assumes that all condensates precipitate immediately (AMS, cited 2020: pseudoadiabatic lapse rate).Reversible:Reversible moist-adiabat, which assumes that all condensates remain in the rising parcel (AMS, cited 2020: reversible moist-adiabatic process).

Table S4 .
P-values of the T-test for the null hypothesis that the difference in mean overpre- diction averaged over 10 • N/S and averaged only over regions of strong mean ascent (ω500 < −35 hPa/d, indicated with an asterisk below) are indistinguishable.The mean difference and the 5-95% confidence interval are also shown.The difference is statistically significant for model configurations that have zonally-asymmetric circulations.(p-value < 5%, indicated in bold).

Table S5 .
P-values of the T-test for the null hypothesis that the difference in mean overprediction between the combined surface warming plus the direct CO2 response and only the surface warming response are indistinguishable.The mean difference and the 5-95% confidence interval are also shown.The difference is statistically significant for all model configurations (p-value < 5%, indicated in bold).

Table S6 .
Same as TableS3but overprediction is evaluated only over regions of strong mean ascent (ω500 < −35 hPa/d, indicated by an asterisk).This filter is not applied to GFDLrce4K as the RCE configuration lacks a climatological large-scale circulation.