1. Introduction
The question of what determines the extratropical tropopause height is of fundamental importance to the general circulation of the atmosphere. It is generally believed that the height of the tropopause is controlled by both the radiative constraint from the stratosphere and the dynamical constraint stemming from the dry baroclinic instability in the tropospheric midlatitudes (Held 1982). Recent studies have also indicated the importance of the stratospheric large-scale dynamics (e.g., Birner 2010) and the tropospheric moist dynamics (Juckes 2000; Frierson et al. 2006; Frierson 2008; Korty and Schneider 2007; Schneider and O’Gorman 2008; Frierson and Davis 2011; Czaja and Blunt 2011) in regulating the tropopause.
This moisture–tropopause relationship as in Eq. (1), in fact, indicates that it is the large and rare fluctuations of low-level equivalent potential temperature that are able to rise to the tropopause level and further modulate the tropopause potential temperature. In general, it is expected that the larger the fluctuation of low-level 
In this paper we extend the diagnosis of the dynamical relationship between the extratropical tropopause potential temperature and the near-surface equivalent potential temperature distribution to an ensemble of coupled climate models that participated in the Coupled Model Intercomparison Project phase 5 (CMIP5). In particular, we aim to explore whether the dynamical relationship works for CMIP5 models and whether the low-level equivalent potential temperature distribution is able to capture the annual cycle of the extratropical tropopause.
It was recognized back to the IPCC First Assessment Report in 1990 that general circulation models tended to systematically simulate a colder temperature than that of observations and the cold temperature bias was most pronounced in the upper troposphere poleward of 50° latitude in both hemispheres and, to a lesser extent, in tropical and midlatitude lower troposphere (Houghton et al. 1990; Boer et al. 1992). This problem of the general coldness of climate models still remains in the state-of-the-art models that participated in the Fourth and Fifth Assessment Reports [e.g., see Fig. 1 of John and Soden (2007), Fig. 4 of Reichler and Kim (2008), and Fig. 2 of Charlton-Perez et al. (2013)]. However, the underlying reasons for this cold bias remain elusive and possible mechanisms have been proposed such as deficiencies in model physics and vertical resolution. A theoretical explanation was raised by Johnson (1997) from the perspective of entropy balance. Johnson (1997) argued that, in order to simulate a climate state without drift, positive definite nonphysical entropy sources introduced by numerical dispersion/diffusion and other reasons have to be offset through increased infrared cooling, which was believed to cause the “general coldness” in model simulations. It was also suggested in Johnson (1997) that this problem of cold biases could be eliminated in models of isentropic coordinates where nonphysical sources of entropy through numerical diffusion vanish. Studies such as Schaack et al. (2004) and Chen and Rasch (2012) used hybrid isentropic coordinates and found somewhat reduced cold biases in temperature in the upper troposphere and lower stratosphere. However, it is worth noticing that the cold biases in these studies largely remained, which suggests that other factors might also matter. This problem of the general coldness of climate models has a lot of consequences and, for example, is associated with biases in simulated atmospheric general circulation. Equatorward biases exist in the climatological jet position across different models; even worse, they could further affect the extent of the jet shift to external forcings in the future climate. As found in Kidston and Gerber (2010) and Son et al. (2010), in general, models of a more equatorward located climatological jet tend to move farther poleward in the late twenty-first century, which creates large uncertainties in the future projections of the jet shift. Therefore, it is important to better understand the underlying mechanisms of the cold biases in climate models. In this paper, from the perspective of the dynamical relationship in Eq. (1), we will discuss the possible dynamical linkage between the upper- and lower-level cold biases across CMIP5 models.
In this paper, we examine the annual cycle of the extratropical tropopause in an ensemble of CMIP5 models and how it is related to that of the near-surface equivalent potential temperature distribution. Biases, in comparison to reanalyses, will be discussed. This paper is organized as follows. Section 2 describes the reanalysis data and CMIP5 simulations used in this study. In section 3, the links between the annual cycle of the extratropical tropopause and that of the near-surface equivalent potential temperature distribution are discussed. Section 4 concludes the paper.
2. CMIP5 climate models
We make use of an ensemble of the latest generation of the coupled climate models that participated in phase 5 the Coupled Model Intercomparison Project phase 5 (Taylor et al. 2012). In this study, 27 coupled climate models from 17 modeling centers are used based on the availability of daily temperature and daily specific humidity. These models as well as their developing institutes and atmospheric model resolutions are listed in Table 1. Since the daily output of CMIP5 archive is only available on eight pressure levels (1000, 850, 700, 500, 250, 100, 50, and 10 mb), for the calculation of the near-surface 
CMIP5 models used in this study with information on host institute and atmospheric model resolution (L refers to number of vertical levels, T to triangular truncation, and C to cubed sphere).
As a reference, we make use of three reanalyses including the ERA-Interim reanalysis (Dee et al. 2011), the NCEP–DOE Reanalysis II (NCEP2; Kanamitsu et al. 2002), and the NCEP Climate Forecast System Reanalysis (CFSR; Saha et al. 2010). As shown in Wu and Pauluis (2014), these three reanalyses provide rather consistent results on both the annual cycle and interannual variability of the extratropical tropopause. Biases in CMIP5 integrations are identified as the difference between model integrations and the above three reanalyses.
3. Results in CMIP5 models
a. Annual cycle of extratropical tropopause and low-level moisture
As shown in Wu and Pauluis (2014), there is a one-to-one relationship between the extratropical tropopause potential temperature and the near-surface equivalent potential temperature distribution for all seasons and for both the two hemispheres. In other words, a large fluctuation of near-surface 
First of all, Fig. 1a shows the moisture–tropopause relationship in CMIP5 multimodel averages in the NH. In comparison to the reanalysis datasets, the CMIP5 multimodel mean is able to successfully reproduce the one-to-one relationship between the low-level equivalent potential temperature fluctuation and tropopause potential temperature with a close to unity correlation and linear regression coefficient (or slope). However, although the close-to-unity correlation is a robust feature among individual CMIP5 models, there is quite a spread in the modeled slope of the annual cycle (see the results of individual models in supplementary Figs. S1–S3, file JCLI-D-14-00543s1). This is also true for the SH (see Fig. 2a and Figs. S4–S6).
The seasonal cycle of the dynamical relationship between 
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
As in Fig. 1, but for the Southern Hemisphere with 
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
Figure 3 shows the modeled slope for the 27 CMIP5 models and for both the two hemispheres. As mentioned above, the slope of the annual cycle varies a lot from model to model—the NH slope ranges from 0.6 to 1.05 while the SH slope covers from 0.5 to 0.85. The modeled slopes of the two hemispheres are slightly correlated (with a correlation of 0.39, which is statistically significant at the 95% confidence level), which suggests that models that do poorly in one hemisphere tend to perform poorly in the other hemisphere as well. As mentioned in section 2, we group together three reanalysis datasets including ERA-Interim, NCEP2, and CFSR. The confidence interval is constructed by using the bootstrap method, which independently resamples the results with replacement, each time a new slope is calculated using the new samples, and repeating for a large number of times. As shown in Fig. 3, the confidence interval is calculated as the 2.5th and 97.5th percentiles of these new slopes from each resampling (similarly for the confidence intervals in other figures). Therefore, depending on how the modeled slope is compared to the constructed confidence interval, the 27 CMIP5 models can be divided into three groups that have smaller, similar, and larger slopes, all statistically significant at the 95% confidence level.
The CMIP5 slopes of the annual cycle of the moisture–tropopause relationship for the Northern and Southern Hemispheres. The reanalyses are plotted in a thick square with the error bars showing the confidence intervals (see text for more details). The model results are plotted in thin squares with the numbers indicating the model numbers as in Table 1 and the multimodel mean is plotted as a thick gray square.
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
For the NH, group N1 has 12 CMIP5 models that have smaller linear regression coefficients than that of the reanalyses, group N2 is characterized by a similar annual cycle slope to that of the reanalyses and includes 14 models, and only one model has a larger linear regression coefficient and is included in group N3. A list of the models in N1, N2, and N3 is given in Table 2. Similarly, for the SH, group S1 includes 16 models with smaller linear regression coefficients, and the other 11 models have similar slopes and are included in group S2 (Table 2). Figures 1b–d show the moisture–tropopause relationship for groups N1, N2, and N3, respectively; Figs. 2b and 2c do the same for S1 and S2.
CMIP5 groups: N1, N2, and N3, with the modeled slope of the Northern Hemisphere moisture–tropopause annual cycle smaller, similar, and larger compared to reanalyses, respectively; similarly for S1 and S2. The numbers within the parentheses indicate the models belonging to that group and are sorted out on the order of ascending slope values. See Table 1 for a list of the models.
In addition to the slope of the annual cycle, another striking feature in CMIP5 model simulations is the systematic cold bias in both the near-surface equivalent potential temperature fluctuation and the tropopause potential temperature. This will be further discussed in the next subsection.
The slope of the annual cycle is an important measure of the one-to-one moisture–tropopause relationship. However, as indicated in Fig. 3, nearly half of the models underestimate the slope of the moisture–tropopause annual cycle. In fact, the largest underestimation of the extratropical tropopause occurs in summer, which contributes to the underestimation of the annual cycle slope. For example, the underestimation of the slope in group N1 is largely because of the smaller near-surface 
With these many climate models and distinct representations of moist processes, it is difficult to conclude what exactly is problematic in groups N1 and N3. However, we believe that these discrepancies arise in part due to the inadequate dynamics or physics in climate models. While the ability of cumulus parameterization has been recognized as a significant challenge for the modeling of the tropical climate, our study suggests that similar deficiencies in the representation of moist processes also negatively impact the higher latitudes. The dynamical relationship between the surface and the tropopause could offer a straightforward approach to diagnose such issues in a range of climate models. More detailed sensitivity experiments are needed for a thorough understanding of the mechanisms and we leave that for future work.
b. Systematic “cold” biases
As shown in Figs. 1 and 2, coupled climate models tend to produce systematic cold biases in both the near-surface equivalent potential temperature distribution and the upper-level potential temperature in the extratropics for all seasons and for both hemispheres. This is consistent with the phenomenon of the general coldness of climate models, which is a long-standing problem since the IPCC First Assessment Report in 1990. Johnson (1997) suggested that the general coldness arises from numerical dispersion–diffusion and resulting positive definite nonphysical entropy sources, and thus is likely intrinsic to climate models. Efforts were made using other model coordinates, and cold biases in upper-level temperature were, to some extent, reduced but still retained (e.g., Schaack et al. 2004; Chen and Rasch 2012). Here we further look into the cold biases in the upper-level potential temperature across CMIP5 models and investigate their possible linkage to the cold biases in the near-surface equivalent potential temperature distribution.
Here we focus on the annual averages in NH extratropics, and the intermodel spread as well as the result from reanalyses is shown in Fig. 4. It can be seen that the majority of the CMIP5 models tends to produce cold biases at both the upper and lower levels. Here the term “cold biases” refers to, in comparison to that of reanalyses, smaller 
The Northern Hemisphere annual mean 
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
Although the focus of this study is the overall performance of CMIP5 models, it is probably worth noticing that, for NH annual averages, two of the farthest outliers are the IPSL-CM5A-LR (model 19) and the IPSL-CM5B-LR (model 21), which are from the same modeling center. For the annual cycle of the NH extratropical tropopause, the IPSL-CM5B-LR performs quite differently from the IPSL-CM5A-LR, and the former has a smaller coefficient of linear regression and deviates farther away from the reanalyses (see Fig. S3). In comparison to IPSL-CM5A-LR, the IPSL-CM5B-LR includes a new version of the physical package and boundary layer parameterization as well as a modified deep convection scheme (Hourdin et al. 2013). As a result, improvements are found in this new version model in the better representation of the convective boundary layer, the cumulus clouds, the diurnal cycle of deep convection over continents, and a Madden–Julian oscillation (MJO)–like signal in the tropics. However, as also demonstrated in Hourdin et al. (2013), significant biases still remain and some are even amplified in this new model version such as a stronger cold bias in tropospheric temperature and a more equatorward located jet stream. This is consistent with what we find here: despite a small improvement in the low-level equivalent potential temperature distribution, the extratropical tropopause potential temperature is even colder in IPSL-CM5B-LR (i.e., as shown in Fig. 4, 
Figure 5a further examines the cold biases in the near-surface equivalent potential temperature distribution and separates that into the contributions from time mean and eddy biases. There is almost no correlation between the simulation of the mean state and that of the eddies across models (the correlation is about 0.16 and is not statistically significant at the 95% confidence level). While CMIP5 multimodel averages can produce more or less similar values of standard deviations of equivalent potential temperature, most of the models systematically underestimate the time mean values of equivalent potential temperature. Figure 5b further attributes the cold biases in mean 
(a) The Northern Hemisphere annual mean 
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
We notice that the cold biases in zonal mean temperature are more prominent in the polar lower stratosphere, as can be found in Fig. 1 of John and Soden (2007), Fig. 4 of Reichler and Kim (2008), and Fig. 2 of Charlton-Perez et al. (2013). But since the maxima of cold biases are located above the tropopause level, we speculate that they are not directly related to near-surface biases.
The SH annual averages across CMIP5 models are slightly different from the NH and the intermodel spread is less organized (not shown). In particular, the cold biases at the upper and lower troposphere are less correlated with a correlation coefficient of 0.38. In the next subsection, we will discuss more about the different behaviors in the two hemispheres.
c. Hemispheric asymmetry in summertime extratropical tropopause and low-level moisture
The summer temperature is higher in the NH than in the SH due to the asymmetric distribution of continents. During the summer months, land temperature increases more rapidly than the ocean temperature due to the lower heat capacity of land. This warming is transferred to the entire atmospheric column; as a result, the tropopause potential temperature is higher in northern summer than in southern summer. This asymmetry between the two summers can be seen in the reanalyses shown in Fig. 6a; the northern summer is about 10 K warmer at both the upper and lower troposphere than the southern summer.
(a) The dynamical relationship for NH summer (indicated by red circles) and for SH summer (indicated by blue crosses) for CMIP5 models. (b) The difference between NH and SH summer for CMIP5 models (thin empty squares). The results for the reanalyses are shown as a reference with a thick black square. The CMIP5 multimodel average is plotted in thick red and thick blue in (a) and by a thick gray square in (b).
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
Here we only estimate the near-surface equivalent potential temperature and the extratropical tropopause potential temperature at the 25°–35° and 35°–45° latitude bands, respectively, but the large asymmetry is also true for the whole hemispheric average and the northern summer is warmer because of the greater land fraction in the NH (see Figs. 1 and 2 of Kang et al. 2015). In fact, the warmer northern summer further leads to a warmer NH in the annual average than the SH, which potentially has important implications for the position of the intertropical convergence zone and the tropical rainfall belt (Kang et al. 2008). Therefore, it is important that climate models produce the right amount of hemispheric asymmetry.
Figure 6a shows the dynamical relationship in northern summer averages and in southern summer averages across 27 CMIP5 models. As can be seen, in northern summer, the majority of the CMIP5 models underestimate both the tropopause potential temperature and the near-surface distribution of equivalent potential temperature, which is known as the general coldness of climate models. In fact, the largest cold biases in multimodel averages occur in northern summer. In southern summer, while a large number of models also underestimate the tropopause potential temperature, the simulation of near-surface equivalent potential temperature distribution across models is rather scattered.
Figure 6b shows the difference between northern summer and southern summer for reanalyses and models. By taking the difference between the two summers, one can remove the global cold bias and better capture the difference in annual cycle over land and ocean. It can be seen that a large part of models underestimates the asymmetry between the two summers, by about 3 K at the lower level and about 2 K at the upper level in multimodel averages.
To further examine the lack of asymmetry at lower levels, Fig. 7a separates that into the contributions from time mean and eddy components of equivalent potential temperature. It is found that it is mainly the underestimation of the mean 
The difference between NH JJA and SH DJF in (a) time-mean 
Citation: Journal of Climate 28, 12; 10.1175/JCLI-D-14-00543.1
Therefore, it is found here that in reanalyses a large asymmetry exists in both the upper and lower troposphere, with the northern summer about 10 K warmer than the southern summer. However, coupled climate models systematically underestimate this hemispheric asymmetry by about 3–4 K. This lack of asymmetry at lower levels largely comes from the fact that the simulated northern summer is too dry in time-mean specific humidity, which reduces the low-level fluctuations of moisture. This underestimation of low-level moisture in northern summer is further related to the upper-level potential temperature via moist dynamical processes, and as a result the simulated extratropical tropopause is too cold in northern summer relative to southern summer, leading to an underestimation of hemispheric asymmetry in extratropical tropopause potential temperature. Therefore, a model’s incapability to reproduce the summer asymmetry is often tied to its incapability to capture the large equivalent potential temperature during northern summer.
4. Discussion and conclusions
This study diagnoses the dynamical relationship that connects the extratropical tropopause potential temperature to the near-surface equivalent potential temperature distribution using an ensemble of CMIP5 coupled climate models. This moisture–tropopause relationship, in fact, pictures the midlatitude moist processes that carry the subtropical low-level poleward-moving air parcels upward and poleward to the extratropical tropopause. As in Wu and Pauluis (2014), a one-to-one relationship was found between the near-surface equivalent potential temperature distribution and the extratropical tropopause potential temperature for the annual cycle, which is a robust feature among different reanalyses. The annual cycle is characterized by a correlation coefficient very close to one and also a slope close to one, which is above 0.8 for the NH and above 0.7 for the SH. In this study, with 27 climate models from the CMIP5 archive, we explore the representation of the extratropical tropopause annual cycle, and in particular examine whether these state-of-the-art models are able to capture the one-to-one relationship between the upper and lower levels. For reference, three reanalyses (ERA-Interim, NCEP2, and CFSR) are used.
Here we summarize the findings:
- In general, CMIP5 multimodel averages are able to produce the one-to-one dynamical relationship between the near-surface equivalent potential temperature distribution and the extratropical tropopause potential temperature for both the Northern and Southern Hemispheres. The correlation coefficient is very close to one and the linear regression coefficient is largely similar to that of the reanalyses. However, “cold” biases are seen at both the upper and lower levels and are universal for all seasons and for both the two hemispheres, systematically for all CMIP5 models. This general coldness of climate models is a long-standing issue dated back to the IPCC First Assessment Report in 1990 and still remains.
- Looking into individual models, a large intermodel spread is found and a large part of CMIP5 models underestimates the slope of the dynamical relationship for the annual cycle. The smaller slope is mostly due to the underestimation of the extratropical tropopause potential temperature in northern (NH) summer [June–August (JJA)] and in southern (SH) summer [December–February (DJF)]. This indicates that, in some model simulations, even with similar values of equivalent potential temperature, the low-level air parcels are not able to rise to the extratropical tropopause level. This might suggest possible issues regarding the representation of the moist processes in the subtropical and midlatitude regions in some models.
- The systematic cold biases in CMIP5 models are further investigated, in particular in Northern Hemisphere annual averages. It is found that the cold biases in near-surface equivalent potential temperature and in extratropical tropopause potential temperature are correlated across the 27 CMIP5 models. In general, models with a colder bias at the lower level tend to have a colder bias at the upper level as well. In addition, the cold biases in near-surface equivalent potential temperature distribution are largely a result of cold biases in temperature and dry biases in specific humidity at lower levels. It is noted here that, in general, models with a finer horizontal resolution as a whole appear to have smaller cold biases at both the upper and lower levels than those with a coarser resolution.
- As mentioned above, the underestimation of the annual cycle is largely due to the poor representations of the northern summer and the southern summer. While the reanalyses show a large asymmetry between the two summers with about 10 K larger values of near-surface equivalent potential temperature and extratropical tropopause potential temperature in northern summer, a large part of models fails to produce the hemispheric asymmetry by about 3–4 K. In comparison to southern summer, the northern summer is found to be too dry in mean specific humidity, which leads to reduced fluctuations of low-level equivalent potential temperature and extratropical tropopause potential temperature.
The annual cycle of the extratropical tropopause is largely dominated by the near-surface mean equivalent potential temperature, which can be partially understood from radiative constraints as in previous studies (e.g., Held 1982; Thuburn and Craig 2000; Schneider 2007). However, the fact that the relationship (1) relates the surface equivalent potential temperature to the extratropical tropopause temperature emphasizes the importance of moist processes for the maintenance of the extratropical tropopause. The contribution from the eddy component, is however also significant, especially in northern summer, and will be discussed in a follow-up paper.
This study applies the dynamical relationship proposed in Wu and Pauluis (2014) to an ensemble of CMIP5 models—in particular, the representation of the annual cycle. The good correlation in both reanalyses and CMIP5 models, as seen in Figs. 1 and 2, is largely due to the dominance of the annual cycle. In the annual cycle, links between the upper and lower troposphere are also seen in model simulations and they might, in fact, suggest possible solutions to the deficiencies of model simulations. For example, as for the problem of the general coldness of climate models, perhaps a finer horizontal resolution or/and a better representation of the boundary layer temperature and humidity distribution might help reduce the cold biases at upper troposphere lower stratosphere. In addition, we believe that the diagnosis using the dynamical relationship is a good and easy way to examine the subtropical and midlatitude moist processes in a group of climate models, and in particular to explore whether the moist convection schemes or large-scale dynamics are successful or not in representing the moist processes. More parameter sensitivity experiments are needed to further explore how the dynamical relationship varies with parameters in the moist convection schemes. This will help better interpret the CMIP5 results and will lead to an improved understanding and representation of moist dynamical processes.
The authors thank Prof. Tapio Schneider and two anonymous reviewers for constructive comments on the manuscript. We are also thankful to the useful comments from Prof. David Raymond during the 19th Conference on Atmospheric and Oceanic Fluid Dynamics. We also acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 1 of this paper) for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. YW and OMP have been supported by the NSF under Grant AGS-0944058 for this work. YW also acknowledge the support from a startup fund from the Department of Earth, Atmospheric, and Planetary Sciences at Purdue University. OMP has also been supported by the NYU Abu Dhabi Institute under Grant G1102.
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