## 1. Introduction

Atmospheric moist convection, through the associated latent heat release and vertical eddy transport of heat, moisture, momentum, and chemical species, plays a key role in global atmospheric circulation, extreme weather systems, climate variability, and climate change (e.g., Houze 2014; Emanuel 1994). Starting from the 1940s, many field experiments have been conducted to study the atmospheric moist convection, such as the Thunderstorm Project (Byers et al. 1946), the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE; Kuettner 1974), the Monsoon Experiment (MONEX; Fein and Kuettner 1980), the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE; Webster and Lukas 1992), the Year of Tropical Convection (YOTC; Waliser et al. 2012), and the Cooperative Indian Ocean Experiment on Intraseasonal Variability in the Year 2011 (CINDY2011; Zhang et al. 2013). Satellite observations, such as those conducted by the International Satellite Cloud Climatology Project (ISCCP; Schiffer and Rossow 1983) and the Tropical Rainfall Measuring Mission (TRMM; Simpson et al. 1988), also provided the large-scale climatology and variability of atmospheric moist convection.

The major forms of atmospheric moist convection include shallow and midtop convection, isolated thunderstorms, and organized mesoscale convective systems (MCSs) (see Houze 2014). The MCSs contribute significantly to the total precipitation and latent heating in the tropics. An MCS generally contains both convective precipitation and stratiform precipitation regions, and the two types of precipitation are associated with significantly different vertical heating and moistening profiles (e.g., Houze 1982; Mapes and Houze 1995). The developing stage of an MCS is often associated with premoistening of the lower troposphere by shallow and midtop convection (e.g., Johnson and Lin 1997; Johnson et al. 1999; Mapes and Lin 2005; Mapes et al. 2006). The mature and decaying stages of an MCS usually generate downdrafts, with the unsaturated convective downdrafts cooling and drying the boundary layer while the mesoscale downdrafts drying the lower troposphere, both of which tend to suppress future development of new convection (e.g., Zipser 1969, 1977; Houze 1977, 1982).

Because the horizontal scale of atmospheric moist convection is generally smaller than the grid size of global climate models, it has to be represented by physical parameterization schemes in the models, which are often called moist convection schemes. The task of a moist convection scheme is to predict the intensity and vertical distribution of convective fluxes. A moist convection scheme generally has two components: the cloud model and the closure assumption. Although the two components are interacting with each other, the cloud model affects more the vertical distribution of convective fluxes, while the closure assumption affects more the total intensity of convective fluxes. Because the closure assumption determines when the convection will happen and how strong the convective fluxes will be, it is generally considered as a more fundamental characteristic of a convection scheme. The closure assumptions used in the first generation of moist convection schemes developed in the 1960s through 1980s can be divided into two groups: those based on the moisture convergence (e.g., Kuo 1965, 1974; Krishnamurti et al. 1976; Anthes 1977; Molinari 1985; Bougeault 1985; Tiedtke 1989) and those based on convective instability (e.g., Manabe et al. 1965; Arakawa and Schubert 1974; Betts 1986). Then in the late 1980s and early 1990s, the moisture convergence closures were criticized seriously and started to fade from the global climate models (Emanuel et al. 1994). Most of the models adopted the convective instability closures, many of which were based on the assumption of convective quasi-equilibrium (CQE) (see Emanuel and Raymond 1993; Smith 1997).

The CQE assumption has been the dominant framework for modeling atmospheric moist convection for at least two decades (Emanuel et al. 1994; Emanuel 2007; Neelin et al. 2008; Raymond and Herman 2011). In this view deep convection develops or decays almost instantly to remove any change of convective instability, which is often represented by convective available potential energy (CAPE) or cloud work function, making the troposphere temperature always tied to the boundary layer moist static energy (MSE). The CQE assumption is very attractive for theoretical modeling because it leads to a very simple picture for global atmospheric circulation and climate variability, ranging from the Hadley and Walker circulations to the Madden–Julian oscillation (Emanuel et al. 1994; Emanuel 2007).

There have been many implementations of the CQE hypothesis, which can be categorized into two different types: the flux type and the state type (Fig. 1). They are basically two different ways to decompose and constrain the change of CAPE or the cloud work function. The flux-type CQE decomposes the CAPE change into its large-scale component and convective component, and requires that the CAPE change is much smaller than any of the two flux terms. It was first proposed for the full troposphere (Arakawa and Schubert 1974; Moorthi and Suarez 1992; Randall and Pan 1993; Zhang and McFarlane 1995) and later also applied to only the boundary layer (Raymond 1995; Emanuel 1995). There is also a variant of the flux-type CQE called free tropospheric CQE by Zhang (2002) and environmental CQE by Bechtold et al. (2014), which is applied only to the free troposphere and tends to decouple the free troposphere from the boundary layer. Observational budget analysis showed that the flux-type CQE generally is not valid at hourly time scales, but becomes valid at daily and longer time scales (Arakawa and Schubert 1974; Zhang 2003; Donner and Phillips 2003).

It is important to note that in climate model implementations of the flux-type CQE, a relaxation time is often introduced for convective adjustment (e.g., Moorthi and Suarez 1992; Zhang and McFarlane 1995). In this way, the convective instability is not removed instantly, which tends to make the thermodynamic structure of the model atmosphere shift away from the CQE.

The state-type CQE, on the other hand, provides a more strict constraint on CAPE change by decomposing it into its boundary layer component and free troposphere component, and requires that the CAPE change is much smaller than any of the two state change terms. It was first proposed for the full troposphere (Manabe et al. 1965; Betts 1986; Emanuel et al. 1994) and later also applied to only the lower troposphere (Mapes 2000; Majda and Shefter 2001; Khouider and Majda 2006; Raymond et al. 2007; Kuang 2008). Because the state change terms are generally much smaller than the flux terms, the validity of flux-type CQE does not guarantee the validity of state-type CQE. Brown and Bretherton (1997) conducted the first observational validation using shipborne surface observations and satellite-derived troposphere-mean temperature. They found that the constants of proportionality between boundary layer MSE and troposphere-mean temperature were only half of the CQE-predicted value even when the data were subject to a strict precipitation window and averaged over a large region for a long time period. Analysis of soundings from several field experiments also showed that the CAPE change is dominated by its boundary layer component (Yano et al. 2001; Zhang 2003; Donner and Phillips 2003). These analyses focused on the troposphere-mean temperature, but the validity of the state-type CQE at different vertical levels has not been investigated.

The purpose of this study is to examine the validity of the state-type CQE at different vertical levels using long-term observational data, and compare the simulations of 42 IPCC global climate models with the observations. The questions we address are the following:

- Is the tropical atmosphere in convective quasi-equilibrium? In other words, is the observed relationship between troposphere saturation MSE and boundary layer MSE at different vertical levels consistent with the state-type CQE?
- If the tropical atmosphere behaves differently from the CQE hypothesis, what causes the tropical atmosphere to be far from CQE?
- Could the IPCC global climate models reproduce the observed relationship between troposphere saturation MSE and boundary layer MSE?

This paper is organized as follows. The data and methods used in this study are described in section 2. The observational evaluation of state-type CQE is presented in section 3. The reason why the tropical atmosphere deviates from the state-type CQE is explored in section 4. Comparison of global climate models with observation is reported in section 5. A summary and discussion are given in section 6.

## 2. Data and methods

The data we used are the sounding measurements from 10 tropical stations over tropical convection centers (Fig. 2, Table 1), including three stations from the Atmospheric Radiation Measurement Program (ARM) and seven stations from the Global Telecommunications System (GTS). The soundings were generally launched twice daily, and the time period varies from station to station, ranging from 10 to 37 years. Careful quality control was applied to remove any erroneous report that deviates from the climatological mean at the corresponding station and vertical level by more than three standard deviations. Altogether 103 906 soundings were used in our analysis. When averaged into monthly data, only the months with more than half month of measurements were used, resulting in 2408 months in total. All soundings were averaged over each vertical 25-mb bin, and the properties of boundary layer air were defined as the average of the lowest 50 mb (1 mb = 1 hPa). Thermodynamic variables were calculated following Emanuel (1994). All soundings from the ARM stations have a high resolution of 2 s, which were averaged to a vertical resolution of 25 mb. Soundings from the GTS stations are reported at standard and significant levels, which were interpolated to a vertical resolution of 25 mb. There is one ARM station (Darwin_{ARM}) and one GTS station (Darwin_{GTS}) right next to each other; as will be shown shortly, they gave nearly identical results, suggesting that it is sufficient to use standard and significant levels from the GTS soundings. The sounding measurements are point measurements. However, the results from the two stations at Chukk and Ponape, which are about 800 km apart, are nearly identical, suggesting that the statistical properties of the soundings represent the properties of a much larger region. This is consistent with the results of previous sounding analysis (Holloway and Neelin 2007).

Tropical sounding stations used in this study.

We also analyzed the simulations of 42 global climate models participating in phases 3 and 5 of the Coupled Model Intercomparison Project (CMIP3 and CMIP5) that are associated with the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4) and Fifth Assessment Report (AR5), respectively (Tables 2 and 3). We used 20 years (model years 1979–99) of the historical simulations from 23 CMIP3 models and 19 CMIP5 models. We have tested the effect of the record length by using 110 years of model outputs for some of the models and obtained similar results. Comparison with observations was done for every station except Aero because for that station several models show significant biases in the boundary layer with large negative correlations and regressions between boundary layer temperature and MSE, which is not directly related to the topic of this study. Therefore the station was excluded to streamline the discussion.

List of the CMIP3 models that participate in this study. Models with same or similar deep convection schemes are listed together. For expansions of model name acronyms, see http://www.ametsoc.org/PubsAcronymList.

As in Table 2, but for CMIP5 models. Models with same or similar deep convection schemes are listed together.

## 3. Is the tropical atmosphere in convective quasi-equilibrium?

*c*between two variables

*x*and

*y*is defined aswhere

*x*

_{i}is the

*x*value for observation

*i*,

*X*is the mean

*x*value,

*y*

_{i}is the

*y*value for observation

*i*, and

*Y*is the mean

*y*value. The linear regression coefficient

*r*between

*y*and

*x*is defined as

The state-type CQE predicts a value of unity for both the regression and correlation coefficients. Some implementations of the state-type CQE involved a small phase lag of several hours between the boundary layer MSE and the troposphere saturation MSE (Emanuel 1993; Neelin and Yu 1994), but when the two variables are averaged over a long time period (e.g., one week and up), their regression and correlation coefficients would still be close to unity. Here we use the reversible adiabat. If pseudoadiabat is used, the CQE-predicted regression coefficient would be larger than unity. Figure 3 demonstrates three key points. First and most importantly, the tropical atmosphere is quite close to CQE in the lower troposphere below 800 mb, but deviates considerably from CQE in the middle and upper troposphere with much weaker troposphere warming (cooling) associated with an increase (decrease) of boundary layer MSE. This is true for both the raw soundings and the monthly means. Although the regression coefficients are significantly larger for monthly means, they are still much less than unity in the middle and upper troposphere. Second, there is no significant difference between ocean and land stations except for monthly means in the upper troposphere, where ocean stations show larger regression coefficients than land stations. Third, although the regression coefficients decrease quickly with height, the correlation coefficients are highly significant throughout the troposphere, and do not decrease as quickly with height especially for the monthly data. These suggest that the small regression coefficients in the upper troposphere are contributed by some well-defined signals that are highly correlated with the boundary layer MSE but with regression coefficients different from those for deep convection. In addition, there is a middle troposphere minimum for both regression and correlation in many stations, which is especially significant for the monthly data, suggesting some vertical structure with a minimum variability in the middle troposphere. We will provide the detailed discussion for these features in the later section.

To give a more quantitative measure of the validity of CQE for different layers of the troposphere, we calculated the regression coefficients between layer-averaged saturation MSE and boundary layer MSE for full troposphere (FT), lower troposphere (LT), middle troposphere (MT), and upper troposphere (UT) for both raw soundings and monthly means (Fig. 4). For the full troposphere, the regression coefficients for all stations are less than 0.27 for raw soundings and less than 0.43 for monthly means, suggesting that the full troposphere CQE is not valid. For the lower troposphere, the regression coefficients are between 0.23 and 0.57 for raw soundings and between 0.24 and 0.68 for monthly means, suggesting that the lower troposphere CQE is a better but not necessarily a very satisfying approximation.

Figures 3 and 4 are the results for hourly and monthly time scales. To examine the validity on longer time scales, we calculated the cross spectrum between monthly layer-averaged saturation MSE and boundary layer MSE (Fig. 5). Also shown are the linear regression coefficients between the two time series filtered at different frequency bands including subannual, annual, biennial, ENSO, and decadal time scales. For all time scales resolved by the observational data, the full-troposphere-averaged saturation MSE is significantly correlated with boundary layer MSE, but the regression coefficients are much less than unity, although they sometimes do increase at longer time scales (Fig. 5). Therefore the full troposphere CQE is not valid at any of these time scales. Moreover, although the regression coefficients in the upper troposphere are much smaller than those in the lower troposphere, the coherencies do not change much and are highly significant, suggesting again that the small regression coefficients in the upper troposphere are contributed by well-defined signals, which will be discussed in the next section.

## 4. What causes the tropical atmosphere to be far from convective quasi-equilibrium?

To understand what causes the tropical atmosphere, especially the middle and upper troposphere to be far from CQE, we discuss the relation between upper troposphere saturation MSE and boundary layer MSE by examining the scatterplot for all 5945 raw soundings at station Manus (Fig. 6). The brown line is the line of linear fit with a slope of 0.14 and a correlation of 0.4. The red line is the reference line having CQE-predicted slope of unity. All the data points can be divided into five regions on the plot as shown by the different colors. Those in the regions labeled phase I and phase III tend to reduce the regression coefficient and move the upper troposphere farther away from the CQE. On the other hand, the data points falling in the regions labeled phase II and phase IV tend to increase the regression coefficient and move the upper troposphere closer to CQE. The data points in the white color regions are close to the current linear fit and would not affect it much. Therefore, we also show in Fig. 6 the averaged temperature anomaly profiles for the four phases at Manus, together with similar profiles for all other stations stratified using the same method. Interestingly, the profiles for each phase are quite consistent among the different stations. The temperature anomaly profiles for phases I, II, III, and IV resemble the typical profiles for shallow convection, deep convection, stratiform precipitation, and very weak or no convection, respectively (Houze 2014), which is also confirmed by the corresponding saturation MSE anomaly profiles (Fig. 7). This suggests that deviation of middle and upper troposphere from CQE might be caused by shallow convection and stratiform precipitation, because they both lead to the reversal of the sign in heating between the upper and lower troposphere (including boundary layer) and thus reduce the regression coefficient between upper troposphere and boundary layer. However, since they are well-defined vertical modes instead of random noise, they do not reduce the correlation in the upper troposphere. In addition, they both have a node in the middle troposphere, and tend to reduce both the regression and correlation there. These processes described above may explain the key features in Fig. 3.

Another physical process that could shift the atmosphere away from CQE is the convective downdraft in deep convection, which often decouples the boundary layer from the free troposphere (e.g., Zipser 1969; Barnes and Garstang 1982). The convective downdrafts in deep convection usually transport low MSE air from the middle troposphere into the boundary layer and therefore significantly decrease the boundary layer MSE. Sometimes the boundary layer MSE becomes so low after deep convection that the CAPE value becomes very negative. Then any small increase of boundary layer MSE will not lead to a positive CAPE or deep convection, and thus will not cause any change in the free troposphere. In this way, the boundary layer is decoupled from the free troposphere, and the CQE hypothesis no longer works.

As shown in many observational studies, the life cycle of an MCS consists of four phases: shallow convection, deep convection, stratiform precipitation, and no convection (Houze 2014; Mapes and Lin 2005). Cloud clusters at larger spatial scales or in longer time scales are ensembles of MCSs and thus also contain all the four phases (Mapes et al. 2006; Straub and Kiladis 2003; Lin et al. 2004). Therefore, the middle and upper troposphere are bound to deviate from CQE because of contributions from shallow convection and stratiform precipitation; convective downdrafts could also decouple the boundary layer from the free troposphere after deep convection. Some previous studies used monthly precipitation window to examine only the months with heavy precipitation, but found that the regression coefficients are still far from the CQE-predicted value (Brown and Bretherton 1997). This may be because the months with heavy precipitation are periods with many MCSs consisting of all the four phases, and the shallow convection and stratiform precipitation may cause the middle and upper troposphere to deviate from the CQE. These are the possible explanations for the failure of the CQE hypothesis at larger spatial scales and longer time scales (Fig. 5) and even for time periods with heavy precipitation (Brown and Bretherton 1997).

## 5. How do the climate models perform?

Next, we analyze the simulations of 42 IPCC CMIP3 and CMIP5 global climate models and compare them with the observations. We used 20 years (1979–99) of monthly outputs from 23 CMIP3 models and 19 CMIP5 models. We have tested the effect of the record length by using 110 years of model outputs for some of the models and obtained similar results. Figure 8 is the same as Fig. 3 except for the comparison with the models, while the corresponding layer averages are shown in Fig. 9. Of the models investigated, 11 CMIP3 models (top panel of Fig. 9, CNRM and the models to its left) and 12 CMIP5 models (bottom panel of Fig. 9, MIR4 and models to its left) produce warming that is comparable to or smaller than the observations for the full troposphere as well as its different layers. The other 12 CMIP3 models and 7 CMIP5 models produce overly strong warming that is closer to CQE than observations for the full troposphere as well as its different layers. Generally, the CMIP5 models reveal noticeable improvement compared with the CMIP3 models by simulating weaker troposphere warming and moving farther away from the unrealistic CQE. This may be due to the recent trend of adding convective triggers in many climate models. However, some CMIP5 models (bottom panel of Fig. 9, BCCC and models to its left) are producing too weak full troposphere warming with unrealistic cooling in the middle and upper troposphere, suggesting that the deep convection may be oversuppressed in those models. A comparison of the model correlation profiles with observation (Fig. 9) shows that many models can reproduce the middle troposphere minimum in correlation, which is very encouraging. Since the instantaneous vertical profiles are not available for the models, we cannot examine the model simulation of the four different phases of convection.

It is very difficult to get the detailed information about the convection schemes used in each model, and the convection scheme is always interacting with other physical parameterization schemes used in the model, such as the boundary layer scheme and microphysics scheme. Model resolution, time step, and even the order for calling different parameterization schemes could also affect the simulation results. We tried to collect the information related to model convection schemes available in journal publications and online documents and have provided it in Tables 2 and 3 (which also contain the labels for model names used herein). Comparison of Fig. 9 with Tables 2 and 3 reveals three points. First, models using state-type CQE closures (CAPE or cloud-base buoyancy) tend to produce larger warming than observation throughout the troposphere (CMIP3 models: GISA, IPSL, MPIC, GISH, GISR, INGV, CSIR, CSI2, and INMC and CMIP5 models: IPSL, IPSM, GISR, MPIE, and GISH), although the Met Office (UKMO) models and the CMIP5 MRIC model tend to produce a weaker warming. Second, models using flux-type CQE closures (CAPE budget) produce a wide range of warming (CMIP3 models: GFDL, GFD1, CCSM, PCM1, CCCM, CC63, IAPC, MIRM, MIRH, and MRIC and CMIP5 models: CCSM, CANE, FGSG, NORE, BCCC, MIRE, MIR4, and MIR5). This may be related to many factors such as the different relaxation time and convective triggers used by the different convection schemes. Third, the only two models using the moisture convergence closure (CNRM and BCCR) produce a warming similar to observation, but the sample is too small to derive any conclusion. It is great to see that several new convection schemes have been developed in the recent years focusing on the improvement of cloud models (e.g., Chikira and Sugiyama 2010; Wu 2012; Yoshimura et al. 2015). More work is needed on improving the closure assumptions and convective triggers.

## 6. Summary and discussion

This study examines the validity of the state-type CQE at different vertical levels using long-term observational data. The results suggest that the tropical atmosphere is far from CQE with much weaker warming in the middle and upper troposphere, which is true for all time scales resolved by the observational data, ranging from hourly to interannual and decadal variability. This is possibly caused by the ubiquitous existence of shallow convection and stratiform precipitation, both leading to the sign reversal of heating from the lower troposphere to the upper troposphere. The simulations by 42 CMIP3 and CMIP5 global climate models are also analyzed and compared with the observations.

The full troposphere CQE is basically a two-phase first-vertical-mode view of the tropical atmosphere with the atmosphere switching between two phases: deep convection and no convection (or weak deep convection), which is an oversimplification of the observed four-phase structure including shallow convection and stratiform precipitation (Fig. 10). This simplification has at least two important impacts on model-simulated global atmospheric circulation and climate variability. First, as shown in our results, the full troposphere CQE assumption tends to significantly overestimate the troposphere heating associated with the MSE increase in the boundary layer. Since the boundary layer MSE is closely connected to sea surface temperature (SST), that bias may contribute to the distorted heating response to SST in many climate models, which is one of the main reasons for the excessive Bjerknes feedback, double-ITCZ problem, and biased interannual variability in those models (e.g., Lin 2007; Zhang et al. 2012). Therefore, the double-ITCZ problem may be alleviated by reducing the overly strong troposphere heating through, for example, better representation of shallow convection and stratiform precipitation. Second, the full troposphere CQE assumption overemphasizes the first-vertical-mode phases but neglects the second-vertical-mode phases. As shown by many observational studies, the two second-vertical-mode phases (shallow convection and stratiform precipitation) play an important role in the Madden–Julian oscillation (MJO) (Lin et al. 2004) and convectively coupled equatorial waves (CCEW) (Straub and Kiladis 2003) by changing the heating structure, prolonging the preconditioning period, and reducing the phase speed. Therefore, better representation of them may help alleviate the MJO and CCEW problems in the models (Lin et al. 2006).

The insightful reviews by two anonymous reviewers and helpful discussions with Chris Bretherton, Kerry Emanuel, and David Neelin are highly appreciated. GTS sounding data were kindly provided by Larry Oolman of University of Wyoming. ARM data are made available through the U.S. Department of Energy as part of the Atmospheric Radiation Measurement Program. We acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI), and the WCRP’s Working Group on Coupled Modelling (WGCM) for their roles in making available the WCRP CMIP3 and CMIP5 multimodel dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy. Jia-Lin Lin was supported by NOAA CPO Grant (GC10-400) under the Modeling, Analysis and Prediction (MAP) Program, the National Aeronautics and Space Administration (NASA) under MAP program and by NSF Grants ATM-0745872 and AGS-1347132. Toshiaki Shinoda is supported by NOAA CPO Grant (GC10-400) under the MAP Program, NSF Grants OCE-0453046, AGS-0966844, ATM-0745897, AGS-1347132, and basic research (6.1) projects sponsored by the Office of Naval Research (ONR) under program element 601153N. Shuanglin Li is supported by the National Key Basic Research and Development (973) Program of China (2012CB417403 and 2015CB453202).

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