Changes in the Tropical Lapse Rate due to Entrainment and Their Impact on Climate Sensitivity

The tropical temperature in the free troposphere deviates from a theoretical moist‐adiabat. The overall deviations are attributed to the entrainment of dry surrounding air. The deviations gradually approach zero in the upper troposphere, which we explain with a buoyancy‐sorting mechanism: the height to which individual convective parcels rise depends on parcel buoyancy, which is closely tied to the impact of entrainment during ascent. In higher altitudes, the temperature is increasingly controlled by the convective parcels that are warmer and more buoyant because of weaker entrainment effects. We represent such temperature deviations from moist‐adiabats in a clear‐sky one‐dimensional radiative‐convective equilibrium model. Compared with a moist‐adiabatic adjustment, having the entrainment‐induced temperature deviations lead to higher clear‐sky climate sensitivity. As the impact of entrainment depends on the saturation deficit, which increases with warming, our model predicts even more amplified surface warming from entrainment in a warmer climate.

2 of 9 throughout the entire free troposphere, the mean convection, consisting of numerous convective parcels, has to be moist-adiabatic. This is an unrealistic idea, because the convection that undergoes moist-adiabatic ascent is extremely rare (Romps & Kuang, 2010). Studies using storm-resolving simulations in the idealized configuration of radiative-convective equilibrium (RCE) show that the tropical temperature tends to deviate from moist-adiabats, because the saturated convective air parcels often mix with the unsaturated environmental air, a process which is referred to as entrainment (Seeley & Romps, 2015;Singh & O'Gorman, 2013. Entrainment reduces the temperature by pushing the convective air parcels away from the original moist-adiabatic trajectories. Similar temperature deviations from moist-adiabats have also been found in Coupled Model Intercomparison Project Phase 5 models (Miyawaki et al., 2020;Zhou & Xie, 2019). These results suggest that the assumption of a moist-adiabatic structure of tropical temperature may be too simplistic.
To what extent does the tropospheric temperature obey the moist-adiabatic LR? The answer is central to understanding some of the fundamental questions of climate change. The vertical structure of atmospheric warming matters for radiation in two ways: it directly controls the thermal emission of an atmospheric layer, and limits the abundance of water vapor through its saturation value which varies with temperature. These are particularly important for the radiative response to warming, which is often quantified by the LR and water vapor feedbacks. For a moist-adiabatic thermal structure, the enhanced tropospheric warming aloft relative to the surface allows more longwave emission to space than would be the case for a constant LR, enabling a cooler surface temperature-a negative feedback. However, this negative LR feedback is largely counteracted by the corresponding increase of water vapor following roughly the Clausius-Clapeyron relation (Soden & Held, 2006). More water vapor reduces emissivity, leading to warmer surface temperatures: a positive feedback. Changes in this subtle balance between the negative LR feedback and positive water vapor feedback can strongly affect the net feedback, leading to contrasting changes in the equilibrium climate sensitivity (ECS), which is defined as the steady-state temperature increase due to a doubling of the atmospheric 2 CO E concentration. It has been shown that the same fractional increase of water vapor at different height alters the net feedback, leading to large changes in ECS (Bourdin et al., 2021;Soden & Held, 2006). Thus, a small departure from the moist-adiabatic structure can potentially have a large impact on the ECS as well.
In this study, we look at the vertical temperature structure and seek to better understand the deviations from moist-adiabats. For simplicity, the moist-adiabatic calculation in this study adopts the pseudo-adiabatic formula as used in Bao and Stevens (2021). We show that these deviations can be explained by different degrees of entrainment, an uncertain parameter controlling the behavior of convective parameterizations in models. Hence, our study allows us to quantify how the specification of entrainment in models may affect their climate sensitivity. The climate sensitivity that we focus on is the clear-sky part of ECS, which we refer to as the clear-sky climate sensitivity ( E ). We first look into the tropical LR by analyzing the data from a global storm-resolving model. Then a simple hypothesis is proposed to explain the variations in the LR by the entrainment of dry environmental air. Based on this hypothesis, we represent the new temperature profile by taking into account the temperature deviations from moist-adiabats in a one-dimensional clearsky RCE model-konrad. Finally, we use this model to quantify the impact of temperature deviations from moist-adiabats (and thereby entrainment) on E .

Tropical Temperature Deviates From a Moist-Adiabat due to the Impact of Entrainment
We start by investigating the tropical temperature structure from a global storm-resolving model-ICOsahedral Non-hydrostatic model (ICON; Hohenegger et al., 2020;Zängl et al., 2015). The model is configured to run at a quasi-uniform horizontal mesh of 2.5 km for 40 days from August 1 in 2016. The data from the last 10 days of the simulations are used in the analysis. The initial conditions are from the global meteorological analysis at a grid spacing of 9.5 km from the European Center for Medium Range Weather Forecasts and the lower boundary conditions are daily observed sea surface temperatures. Details about the model setup are provided by Hohenegger et al. (2020). Here, we focus on the tropical ocean regions over 10 E

S.
We hypothesize that the mean tropical thermal structure reflects the composition of convection happening at different values of boundary-layer equivalent potential temperature ( e E  ) and mixing more or less with the environment while rising up, as the temperature in the free troposphere is not determined by the one or a few strongest (or warmest) convection, but rather by the mean convection, which represents a combined effect from all convection occurring over each height. This can be understood with Figure 2. The tropical atmosphere is composed of numerous convective systems. While most convection can reach a relatively lower altitude, the chance of convection occurring at a higher altitude is smaller, and even less can survive up to the tropopause. In order for an air parcel to reach a relatively higher altitude, this parcel has to maintain its positive buoyancy relative to the surrounding environment. Entraining the environmental air into the updraft would successively prevent the parcel from rising higher. As the height that each convective parcel can reach depends on the parcel buoyancy, the upper troposphere is increasingly dominated by the convective parcels that are warmer and more buoyant. These parcels usually arise in a relatively more humid environment so that they can maintain the positive buoyancy. We refer to this height-dependence of parcel buoyancy as the buoyancy sorting of convection. Thus, above a certain level where most air parcels cease to rise, es E  increases with height as shown in Figure 1b, because only the more buoyant air parcels with larger es E  can continue The moist-adiabatic profiles were calculated based on the domain-mean output from the lowest model level near the surface. Differences are shown for the tropic mean state (red) and the moist regions (colors from yellow to blue correspond to the 90th, 99th, 99.9th, 99.99th, and 99.999th percentile of precipitable water). 4 of 9 rising up. This level appears to coincide with the freezing level, which is a stable layer as observed in Johnson et al. (1999) and tends to inhibit cloud growth and promote cloud detrainment (Stevens et al., 2017).
The buoyancy-sorting mechanism has been around for decades and some pioneering studies tried to represent it in convective parameterizations (Emanuel & Živković Rothman, 1999;Raymond & Blyth, 1986). However, it is rarely used to explain the tropical LR changes. One study by Folkins (2002) noticed that the tropical temperature deviates from a moist-adiabat and explained it with the buoyancy-sorting idea. But his results were limited to the tropical tropopause region. Here, we show that the representation of entrainment can affect the temperature profile through the depth of the troposphere. Recently, Zhou and Xie (2019) proposed that the tropical mean temperature structure could not be represented by one convective plume, but rather by a spectral of plumes with different entrainment rates. They developed a spectral plume model with the buoyancy-sorting mechanism to represent the tropical temperature profile.

Representing the Temperature Deviations From Entrainment in Konrad
We aim to represent the temperature profile that takes into account the deviation from the moist-adiabat in a clear-sky one-dimensional RCE model-konrad Kluft et al., 2019). In konrad, the original convective adjustment assumes that the temperature follows exactly a moist-adiabatic LR, based on the surface temperature calculated by a slab-ocean model. To represent the temperature reduction from the impact of entrainment, we adopt the formula derived from the zero-buoyancy entraining plume model by Singh and O'Gorman (2013). The simulated temperature profiles are compared with the tropical mean profile averaged over the period of 2006-2015 from the ERA5 reanalysis data (Hersbach et al., 2019).
Here, we briefly review the main idea of the zero-buoyancy entraining plume model. The zero-buoyancy entraining plume assumes the cloud buoyancy is small. As the plume is saturated at the environment temperature above the cloud base, this allows us to derive the temperature reduction due to entrainment from the plume moist static energy (MSE) budget:

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Then we expand h u * and h e * , and linearize q v * : where u E T and e E T are the temperature of the undiluted air parcel and the environment, respectively, q vu * is the saturation-specific humidity at the temperature of the air parcel, E T  is the temperature difference between the undiluted air parcel and the environment, and p E c is the isobaric specific heat capacity of the dry air. By combining Equations 2 and 3, we integrate vertically to get the temperature reduction from the impact of entrainment ( where RH E is the environmental relative humidity: RH ve ve  q q / * , and b E z is the height of the cloud base. Equation 4 shows that the temperature reduction from the impact of entrainment depends on the entrainment rate as well as the saturation deficit. Following Romps (2014), RH E is predicted by temperature, which exhibits a C shape and is roughly temperature invariant in the free troposphere (Figure 3a). The results are qualitatively consistent if a constant RH E profile is used. We use a fixed entrainment rate profile defined as   ( ) / z z  0 and 0 E  is the entrainment parameter. Because konrad uses pressure as its vertical coordinate, the variables simulated by konrad are converted from pressure to height assuming hydrostatic balance before they are used in Equation 4.
The temperature deviation term is not computed strictly from the entraining plume model. For simplicity, we utilize the formula (Equation 4) derived from the model and calculate the temperature deviation term directly. The final temperature profile is obtained by subtracting this temperature deviation term from the temperature assuming the moist-adiabatic adjustment. So we first calculate the moist-adiabatic temperature profile based on the surface temperature, and then use this temperature profile to compute q ve * in Equation 4. Although q ve * corresponds to the saturation-specific humidity of the environment, such a simplification would not qualitatively alter the results. Most importantly, the key impact of climate change, that is, the Clausius-Clapeyron increase of q ve * , is captured.
One major issue with the zero-buoyancy entraining plume model is that it assumes that the temperature in the free troposphere is controlled by the mean convection, but fails to represent the buoyancy sorting of convection. As a result, the upper-tropospheric temperature depicted by the model is unrealistic. Here, to doubling. Then, the climate feedback parameter is defined as: Following the method introduced by Gregory et al. (2004), we regress changes in the TOA radiative flux against changes in SST for the daily mean output ( Figure S1). The data over the initial period of stratospheric adjustment are excluded. As konrad is an idealized model in which many processes are either neglected or simplified, the uncertainty in the regression analysis is extremely small. We obtain almost perfect linear relationships between TOA radiative imbalance and SST change. The intercept of the regression line gives the effective radiative forcing ( 2 CO 2 E F   ), and the regression slope is the feedback parameter ( E ). Figure 3b shows the profiles of temperature deviations from the moist-adiabats simulated by konrad. Due to the impact of entrainment, the free troposphere is colder. This cooling effect is increased with a larger entrainment parameter ( We represent such temperature deviations from moist-adiabats in a clear-sky one-dimensional RCE model and quantify its impact on the clear-sky climate sensitivity ( E ). The temperature deviation term is represented by weighting the formula derived from a zero-buoyancy entraining plume model with a height-dependent coefficient. We show that this idealized representation of entrainment is capable of producing temperature profiles more similar to the ERA5 reanalysis. Compared with a strict moist-adiabatic adjustment, having this entrainment-induced temperature deviation leads to higher E , because entrainment alters the LR in a way that more closely resembles a constant LR. This weakens the negative LR feedback. Meanwhile, the positive water vapor feedback changes less due to compensating effects from drying in the upper and lower troposphere. Thus, E  increases because the total feedback change is dominated by the change in the LR feedback. Finally, as the impact of entrainment depends on the saturation deficit, which increases with warming due to the Clausius-Clapeyron relation, this model predicts even more amplified surface warming from entrainment in a warmer climate.

LR Effects on Climate Sensitivity
Although uncertainties in projected warming are largely contributed by the cloud feedback, this study emphasizes the importance of understanding how the clear-sky feedbacks change with warming. The CMIP6 model ensemble is capable of replicating the observed temperature deviations from moist-adiabats. Still, the spread in temperature deviations among individual models can contribute to the E  uncertainty of 0.2 K E  0.3 K.
Entrainment and its impact on LR can potentially influence clouds and circulation, which are not represented by our simple model. Results from idealized RCE simulations show that increased impact of entrainment can lead to more organized convection (Tompkins & Semie, 2017), and climate sensitivity is associated with changes in the degree of organization (Becker & Wing, 2020). A recent observational study showed that deep convective organization modulates tropical radiation budget, which is expected to affect climate sensitivity (Bony et al., 2020). Thus, an improved understanding of the impact of entrainment on climate sensitivity through clouds and circulation is desired.