## Abstract

This paper reports on a new index for low cloud cover (LCC), the estimated cloud-top entrainment index (ECTEI), which is a modification of estimated inversion strength (EIS) and takes into account a cloud-top entrainment (CTE) criterion. Shipboard cloud observation data confirm that the index is strongly correlated with LCC. It is argued here that changes in LCC cannot be fully determined from changes in EIS only, but can be better determined from changes in both EIS and sea surface temperature (SST) based on the ECTEI. Furthermore, it is argued that various proposed predictors of LCC change, including the moist static energy vertical gradient, SST, and midlevel clouds, can be better understood from the perspective of the ECTEI.

## 1. Introduction

Recently, changes in low clouds in a warmer climate and low cloud feedback have been intensively studied (e.g., Blossey et al. 2013; Bretherton and Blossey 2014; Brient and Bony 2013; Kawai 2012; Webb and Lock 2013; Zhang et al. 2013). In addition, there have been a number of studies examining changes in indices of meteorological factors that can be related to low cloud feedback. For example, Brient and Bony (2013) discussed changes in the vertical gradient of moist static energy (MSE) to explain low cloud changes, while Webb et al. (2015) showed that models with few midlevel clouds or low MSE near the top of the boundary layer (BL) tend to have a large positive tropical cloud feedback.

Wood and Bretherton (2006) showed that estimated inversion strength (EIS), which is a modification of lower-tropospheric stability (LTS; Klein and Hartmann 1993), is a physically more plausible and useful index than LTS for determining low cloud cover (LCC) in the present climate (i.e., larger LCC for stronger EIS). This index has been used in many studies. However, Webb et al. (2013) showed that low cloud feedback is positive despite EIS increases in the future climate in almost all models. Qu et al. (2014) proposed that changes in low cloud cover are determined by a linear combination of changes in EIS and sea surface temperature (SST) in the present climate, and that future changes in low cloud cover can be determined from future changes in EIS and SST. Therefore, LCC can decrease even if EIS increases in the future climate, because SST increases significantly.

In this paper, we develop a new index for LCC, the estimated cloud-top entrainment index (ECTEI), which is a modification of EIS and takes into account a cloud-top entrainment (CTE) criterion, using EIS to estimate the temperature gap at the top of the boundary layer [for a preliminary report on this index, see Kawai (2013)]. We argue here that changes in LCC cannot be fully determined by changes in EIS alone, but can be better determined from changes in both EIS and SST based on ECTEI. The relative contributions of changes in EIS and SST can be deduced if we assume that ECTEI determines LCC. LCC changes related to, for instance, the MSE vertical gradient, SST, or midlevel clouds, can also be better understood using ECTEI. The purpose of this paper is to describe the fundamental ideas and relationships deduced from the basic formula for ECTEI.

## 2. A new stability index for low cloud cover

### a. New stability index

#### 1) Concept

The cloud-top entrainment process, which is an important process for low-cloud breakup, has been studied extensively, and the following criterion is widely used (Randall 1980; Deardorff 1980; Kuo and Schubert 1988; Betts and Boers 1990; MacVean and Mason 1990; MacVean 1993; Yamaguchi and Randall 2008; Lock 2009):

where *θ*_{e} is the equivalent potential temperature, *L* is latent heat, *c*_{p} is the specific heat at constant pressure, *q*_{t} is total water specific humidity, and *Δ*_{inv} indicates the gap at the inversion that is defined as the above-inversion value minus the below-inversion value. The factor *k* is constant. MacVean and Mason (1990) derived *k* = 0.70, and this was validated by numerical simulations undertaken by MacVean (1993). Note that while *k* is usually treated as constant, *k* is actually proposed by MacVean and Mason (1990) as a function of temperature and pressure and this dependence is discussed in section 4e. The inequality was developed as a criterion for the situation where a dry air parcel just above a cloud-top inversion can gain negative buoyancy through evaporative cooling due to mixing with the air in a cloud layer. The cloud layer is then broken up by the continuous entrainment of the dry air from above the inversion. The inequality means that a weak temperature inversion and a large humidity gap (i.e., a dry atmosphere just above the inversion) are favorable for causing CTE. Although EIS is known to be a good indicator of LCC, EIS is related only to the inversion strength of (potential) temperature at the cloud top. Therefore, we developed an index that includes information related to the humidity gap, to take into account the CTE criterion.

We roughly assume that *q*_{t} below the inversion is close to *q* near the surface, and *q*_{t} above the inversion is close to the specific humidity *q* at 700 hPa. The *q* difference term is multiplied by a coefficient *C*_{qgap} (≤1.0; e.g., 0.8) because the actual *q*_{t} gap at the inversion is presumably smaller than the *q* difference between the surface and 700 hPa. If potential temperature *θ* is used instead of *θ*_{e} [≈*θ* + (*L*/*c*_{p})*q*] and the cloud water content term is ignored, Eq. (1) can be rewritten as follows:

Here, we use EIS as a substitute for the potential temperature gap, and regard the left-hand side of Eq. (2) as an index (ECTEI) as follows:

where *β* = (1 − *k*)*C*_{qgap}.

ECTEI was developed from the idea of the corrected gap of low-level moist static energy (CGLMSE) index proposed by Kawai and Teixeira (2010).

#### 2) A coefficient in the index

The coefficient *C*_{qgap}, which is the ratio of the *q*_{t} gap at the inversion and the *q* difference between the surface and 700 hPa, must be determined to calculate ECTEI. Radiosonde observation data during a field campaign [East Pacific Investigation of Climate (EPIC); Bretherton et al. 2004] off the coast of Peru were used to obtain the coefficient. All 3-hourly sounding profiles during 13 and 25 October 2001 were used and the number of profiles used in the statistics was 96. The vertical sampling interval of the radiosonde soundings is about 10 m.

An inversion height is determined as a lowest height where the absolute value of the vertical gradient of potential temperature is larger than 0.3 K hPa^{−1}. The gap in *q*_{t} at the inversion is calculated from the specific humidity 100 m below and 150 m above the determined inversion height. Specific humidity near these levels and near 700 hPa are averaged over layers of 50-m thickness to obtain robust values. From this analysis, a value of *C*_{qgap} = 0.76 was obtained. The standard deviation of the sample is 0.186, and the standard error of the mean (i.e., the estimated standard deviation of the sample average) is . Therefore, we use 0.76 as a value of *C*_{qgap} in the following analysis and, together with *k* = 0.70, this results in *β* = 0.23 in Eq. (3).

### b. Observational support

We used the observed cloud fraction derived from shipboard observations, as well as stability indices calculated from reanalysis data, to examine the relationships between LCC and stability indices, including ECTEI. The shipboard observation data for 1957–2002, known as the Extended Edited Cloud Report Archive (EECRA; Hahn and Warren 2009), was used to examine the relationships. Stability indices LTS, EIS, and ECTEI were calculated using the 40-yr ECMWF Re-Analysis (ERA-40) data (Uppala et al. 2005). We regarded the combined cloud cover of stratocumulus, stratus, and sky-obscuring fog as LCC in the present study, as in Koshiro and Shiotani (2014). Figure 1 shows the relationships between the LCC and the stability indices LTS, EIS, and ECTEI calculated simply from both sets of seasonal climatology data, where all the data between 60°N and 60°S for all seasons were used. Figure 1 shows that ECTEI has a very strong correlation with LCC, at least at the same level as EIS, with the correlation coefficients being *R* = 0.32 for LTS, *R* = 0.87 for EIS, and *R* = 0.91 for ECTEI.

Although a value of *β =* 0.23 is used in the calculation of ECTEI, based on *C*_{qgap} = 0.76, which is obtained from radiosonde soundings, and *k* = 0.7 (MacVean and Mason 1990; MacVean 1993), these values have some uncertainties. To examine correlation coefficients between LCC and ECTEI for different values of *β*, correlation coefficients are obtained by sweeping over a range of values for *β* (Fig. 2). Note that ECTEI is equivalent to EIS when *β* = 0 [see Eq. (3)]. Figure 2 shows that the correlation coefficient is highest around *β* = 0.23. In addition, the 95% confidence interval of *β* is 0.22–0.24 when we perform a multiple linear regression analysis using LCC as a dependent variable, and EIS and *q*_{surf} − *q*_{700} as explanatory variables. However, more important and objective information is that the correlation coefficient is relatively high for all values of *β* shown in Fig. 2. At least, we can say that adding the second term in Eq. (3) to EIS does not reduce the strength of the correlation with LCC, and EIS is not the only option, although EIS has been used as an excellent index for LCC. Although we use *β* = 0.23, which is derived from the aforementioned concept together with radiosonde observation, in the following discussion, it should be kept in mind that a wide range of *β* values are consistent with a high correlation with LCC.

## 3. Indices for LCC

### a. Dependency of indices on sea surface temperature

First, we examined the basic dependency of these indices on SST. A profile of a lower troposphere typical of stratocumulus regions was assumed, which had a dry adiabatic lapse rate of 9.8 K km^{−1} below the lifting condensation level (LCL), a moist adiabatic lapse rate inside the cloud layer and free atmosphere, a temperature inversion of 4 K, and relative humidity of 80% at the surface and 30% at 700 hPa. See appendix A for details of the profile used. In addition, we assumed that the SST was 1-K higher than the surface air temperature and that *β* = 0.23. The sensitivities of the results to the profile and the coefficient are discussed later.

Figure 3 shows the dependency of the index differences ECTEI − EIS and LTS − EIS on SST. The mathematical formulations of the index differences, and the qualitative explanation related to the dependency of the differences on SST, are given in appendix B. For a given EIS, LTS is larger (corresponding to more cloud) for a higher SST, and smaller (less cloud) for a lower SST, as is well known. On the other hand, ECTEI is smaller (less cloud) for a higher SST and larger (more cloud) for a lower SST for a given EIS [because of the humidity difference term in Eq. (3)]. Although sensitivity to the profile and the coefficient is discussed in more detail later, sensitivity to relative humidity at 700 hPa only is shown in Fig. 3. The difference between ECTEI and EIS is large for small values of assumed relative humidity at 700 hPa because the second term on the right-hand side of Eq. (3) has a large magnitude. Note that the index difference LTS − EIS does not change with relative humidity at 700 hPa because neither LTS nor EIS takes the humidity profile into account. If ECTEI is a good indicator that is applicable to any time scale of cloud variability, then the dependency of ECTEI − EIS on SST implies that cloud will decrease in the future climate, if EIS does not change or increases by a sufficiently small amount. Climate models suggest that EIS will increase by a small amount with climate warming, but less than LTS (e.g., Qu et al. 2015a; Webb et al. 2013).

### b. LCC dependency on SST

#### 1) Equations

Qu et al. (2014) proposed that a change in LCC can be determined by a linear combination of changes in EIS and SST as follows:

where Δ shows a change in a variable, and (∂LCC/∂EIS)_{SST} and (∂LCC/∂SST)_{EIS} are referred to as the EIS and SST slopes, respectively. A change in ECTEI in Eq. (3) can be written as follows:

Here, it is assumed that air–sea temperature difference and relative humidity at the surface and free troposphere do not change substantially for different SSTs, including SST change due to a climate change. In the present study, we assume that the atmospheric profile in a warmer climate can be calculated as in appendix A using an increased SST. The partial derivative of the second term on the right-hand side of Eq. (5) can be obtained by differentiating the curves in Fig. 3 with respect to SST, and this is shown in Fig. 4 (refer to the left vertical axis). Note that the first term on the right-hand side of Eq. (5) corresponds to a change in temperature inversion (temperature profile) and the second term corresponds to a change in the specific humidity gap (humidity profile).

If LCC is a function of EIS, then

On the other hand, if LCC is a function of ECTEI, ΔLCC can be written as follows, using Eq. (5):

From Eq. (7), the EIS and SST slopes can be written as follows:

The key concept here is that the dependency of LCC on the vertical humidity difference is substituted for approximately by the SST dependency, and this is partly consistent with the discussion in Qu et al. (2015b) and our discussion suggests a quantitative physical justification of their results.

#### 2) Quantitative discussion

First, as for the total derivative of LCC with respect to EIS in Eq. (6), a value for *d*LCC/*d*EIS of 4.7% K^{−1} was obtained from Fig. 1b using the shipboard observation data, as already shown in Koshiro and Shiotani (2014). This value is not greatly dissimilar to the value of 6% K^{−1} shown by Wood and Bretherton (2006).

The EIS slope, (∂LCC/∂EIS)_{SST}, approximately corresponds to *d*LCC/*d*ECTEI from Eq. (8), and this total derivative can be estimated from our analysis in Fig. 1c. This EIS slope was calculated to be 3.1% K^{−1} from the shipboard observations. Qu et al. (2015b) calculated EIS slopes using satellite-based cloud data together with ERA-Interim data. The obtained EIS slopes are 2.6% K^{−1} using the International Satellite Cloud Climatology Project (ISCCP) data, 2.4% K^{−1} using the Pathfinder Atmospheres–Extended (PATMOS-x) data, 3.5% K^{−1} using the Multiangle Imaging SpectroRadiometer (MISR) data, and 3.3% K^{−1} using the Moderate Resolution Imaging Spectroradiometer (MODIS) data. Seethala et al. (2015) derived an EIS slope, where temperature advection is also constant, of about 3.4% and 3.9% K^{−1} based on the ISCCP and PATMOS-x observations, respectively. Our EIS slope is consistent with the slopes from these previous studies whose targets were areas of frequent subtropical stratocumulus occurrence. In addition, Qu et al. (2014) obtained ensemble mean EIS slopes of 1.1% and 1.5% K^{−1} (the intermodel standard deviations are 1.3 and 1.7) using data from phase 3 (CMIP3; Meehl et al. 2007) and phase 5 of the of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012) model simulations, respectively.

The SST slope, (∂LCC/∂SST)_{EIS}, can be approximately calculated from Eq. (9)—that is, as a product of a total derivative *d*LCC/*d*ECTEI and a partial derivative (∂ECTEI/∂SST)_{EIS}. The partial derivative can be obtained from Fig. 4 (left vertical axis), and it is roughly −0.29 K K^{−1} for a typical SST of 295 K and relative humidity at 700 hPa of 30%. Therefore, the SST slope is −0.90% K^{−1} using the same value for *d*LCC/*d*ECTEI based on the observation data as in the discussion above. Qu et al. (2015b) found SST slopes of −2.5% K^{−1} based on ISCCP data, −2.2% K^{−1} based on PATMOS-x data, −1.4% K^{−1} based on MISR data, and −1.0% K^{−1} based on MODIS data. Seethala et al. (2015) calculated SST slopes of about −1.7% and −1.3% K^{−1} based on ISCCP and PATMOS-x observations, respectively. Our SST slope is again consistent with the slopes reported in these studies. Furthermore, we obtained the SST slopes directly from the EECRA shipboard observation data that were used to plot Fig. 1. The EIS was calculated using ERA-40 data, and SST data were taken from the Hadley Centre Sea Ice and Sea Surface Temperature dataset (HadISST) version 1.1 (Rayner et al. 2003), as in Koshiro and Shiotani (2014). Figure 5 shows that low stratiform cloud cover clearly has dependency on SST for given values of EIS. The SST slopes are from −1.1% to −0.6% K^{−1} for each EIS bin shown in Fig. 5. These SST slopes obtained directly from the shipboard observation data are also consistent with the SST slope estimated from the formula of ECTEI (−0.90% K^{−1} at SST = 295 K). As a reference, Qu et al. (2014) obtained SST slopes of −1.1% and −1.2% K^{−1} (the intermodel standard deviations are 1.1 and 1.0) from the CMIP3 and CMIP5 model simulations, respectively. All of these derivatives are summarized in Table 1.

Approximate values of (∂LCC/∂SST)_{EIS} (i.e., SST slopes) are shown on the right vertical axis in Fig. 4 by assuming that *d*LCC/*d*ECTEI = 3.1% K^{−1} for reference. In addition, Figs. 4a–d show the sensitivities of the SST slope to relative humidity at 700 hPa, the temperature gap at the inversion, relative humidity at the surface, and the coefficient *β*, respectively. Figure 4a shows that when relative humidity at 700 hPa is small, the SST slope has a large negative value. It is also clear that the sensitivity of the SST slope to the assumed magnitude of the temperature gap at the inversion (i.e., the vertical temperature profile) is small (Fig. 4b). Figure 4c shows that the magnitude of the sensitivity of the SST slope to the relative humidity at the surface is similar to its sensitivity to relative humidity at 700 hPa. Figure 4c shows that the SST slope has a large negative value for large relative humidity at the surface, in contrast to the sensitivity to relative humidity at 700 hPa. This characteristic can be attributed to the fact that the second term on the right-hand side of Eq. (3) has a large magnitude for a large relative humidity at the surface. The magnitude of the sensitivity of the SST slope to the coefficient *β* is also similar to its sensitivity to relative humidity at 700 hPa (Fig. 4d), where the values *β* = 0.16 and 0.30 correspond to *C*_{qgap} = 0.53 and 1.0 respectively, if *k* is constant as 0.70.

### c. Changes in LCC and vertical difference of MSE

The second term of ECTEI in Eq. (3) includes the vertical difference of the specific humidity, and this inspired us to connect this term with the vertical difference of MSE.

MSE can be written as follows:

The vertical difference in MSE between 700 hPa and the surface, *δ*_{700−surf}MSE, can be written as follows:

Changes can be written as follows:

When temperature in the lower troposphere increases, the magnitude of changes is several times larger in the second term than in the first term on the right-hand side. For instance, when we assume that a change in EIS is negligible under temperature increase of 4 K, the first term and the second term are 1.3 and 5.7 kJ kg^{−1}, respectively, at SST = 295 K (using the atmospheric profiles commonly used in the present study). Therefore, we neglect the first term as follows:

Using the above relationship, we obtain the following equation:

Using Eq. (14),

Equation (15) suggests that the change in LCC is affected by a change in the vertical difference of MSE between 700 hPa and the surface, *δ*_{700−surf}MSE. The coefficient of the second term on the right-hand side, referred to as the *δ*_{700−surf}MSE slope, is 0.71% (kJ kg^{−1})^{−1} for *β* = 0.23 and *d*LCC/*d*ECTEI = 3.1% K^{−1}. These values are similar to the slope of the relationship between the low cloud frequency and the vertical difference of MSE between the inversion and the surface found by Kubar et al. (2011) of 0.74%–2.78% (kJ kg^{−1})^{−1}, although these slopes do not need to be equivalent because the definitions of the vertical difference are not the same. The important point is that our discussion can provide some possible interpretation of the contribution of the change in *δ*_{700−surf}MSE to the change in LCC.

## 4. Discussion

### a. Low cloud cover and the new stability index

In the present study, a new stability index, ECTEI, in which a term using the specific humidity difference between 700 hPa and the surface is added to EIS to take into account a CTE criterion, is proposed. It is worth noting that van der Dussen et al. (2015) showed that the specific humidity difference between 700 hPa and the surface affects the liquid water path (LWP) of stratocumulus using a large-eddy simulation (LES) model. In their simulation, LWP is large for a given LTS when a magnitude of the specific humidity difference between 700 hPa and the surface is small (i.e., the relative humidity at 700 hPa is large). Although their result is related to LWP and our study is related to LCC, their results seem to support the importance of the vertical moisture difference for marine boundary layer clouds.

In section 3, we demonstrated that the SST slope in the present climate obtained by, for example, Qu et al. (2014) and Seethala et al. (2015) and the *δ*_{700−surf}MSE slope calculated by Kubar et al. (2011) can be obtained mathematically, via some approximations, using ECTEI. The essential concept is that the second term (the humidity difference term) of ECTEI is highly correlated with SST and surface MSE via the specific humidity at the surface.

Actually, several studies have found that stratiform low clouds are controlled by meteorological indices other than EIS and SST, including subsidence velocity and horizontal temperature advection (e.g., Myers and Norris 2015, 2016; Seethala et al. 2015). For instance, Myers and Norris (2013) showed that enhanced subsidence promotes reduced LCC for the same value of inversion strength based on satellite cloud data and reanalysis data. Therefore, we cannot conclude that LCC variations are controlled solely by a single index such as EIS or ECTEI. However, the stability indices EIS and LTS are definitely good indices for LCC and have been used in many studies to understand LCC variations. The new ECTEI gives some additional insights for LCC variations by including a humidity difference term. In addition, ECTEI can be useful for interpreting the factors controlling low cloud feedbacks, as described below.

### b. Low cloud feedback and the new stability index

Qu et al. (2014) showed that future changes in LCC can be largely explained by the combination of EIS change and SST change, that is, using Eq. (4). Qu et al. (2015b) and Myers and Norris (2016) suggest that EIS and SST are the most important factors for low cloud feedback, although other factors, including subsidence velocity and horizontal temperature advection, can exert some influence. Brient and Bony (2013) suggested that a decrease in low-level clouds in a warmer climate is caused by an increase in the vertical gradient of MSE, based on a framework of an MSE budget. We consider it useful to interpret low cloud feedback using ECTEI because the concepts described in section 3 suggest a unified physical interpretation of the relationships between low cloud feedback, EIS, SST, and vertical gradients in MSE. However, we should consider the possibility that the EIS and SST slopes, as well as the *δ*_{700−surf}MSE slope obtained in the present study and based on the present climate, may need to be modified to some extent for a discussion related to cloud feedback, as the relationship between LCC and ECTEI might change to some degree under a future climate.

### c. Further insights into low cloud feedback

It is possible that a discussion focused on ECTEI could provide additional insights into low cloud feedback if the relationship between changes in LCC and ECTEI can also be applied to future changes. First, Fig. 4 shows that the magnitude of an SST slope, (∂LCC/∂SST)_{EIS}, is larger for a higher SST. This means that a given SST increase will cause a greater reduction in LCC in a future climate, for example, in the subtropics compared with midlatitudes, because SST is higher in the subtropics.

Second, Fig. 4 shows that the SST slope can be affected by relative humidity at 700 hPa (drier RH_{700} corresponds to larger magnitude of SST slope). This implies that the cloud reduction is greater for low RH_{700} models for a given SST and a given SST increase. The sensitivity of the SST slope to RH_{700} is substantial and the slope varies by a factor of 2 for a range of RH_{700} = 20%–50% (−0.6% K^{−1} for RH_{700} = 50% and −1.0% K^{−1} for RH_{700} = 20% at SST = 295 K, where a *d*LCC/*d*ECTEI value of 3.1% K^{−1} is used). In addition, for comparison, the sensitivity to RH_{700} is comparable to the dependency of the SST slope on SST itself (−0.4% K^{−1} for SST = 280 K and −1.1% K^{−1} for SST = 300 K, where the same value of *d*LCC/*d*ECTEI is used).

Third, the midlevel cloud fraction should have some correlation with RH_{700}, and MSE_{700} should have some correlation with *q*_{700}. Therefore, the SST slope is possibly sensitive to the midlevel cloud fraction and MSE_{700} (models that have a smaller midlevel cloud fraction or smaller MSE_{700} tend to have a larger magnitude SST slope) because the SST slope is sensitive to RH_{700} and *q*_{700}. Hence, the relationships between low cloud feedback and RH_{700}, MSE_{700}, and midlevel clouds (Webb et al. 2015) among CMIP multimodel simulations may be partly explained through the relationship between in LCC and ECTEI.

The second and the third points above are somewhat speculative and are based on the assumption that climate models have relationships between LCC and ECTEI similar to that in observations. However, Qu et al. (2014) showed that the SST and EIS slopes vary widely among models and this implies that changes in LCC and ECTEI also vary widely. Therefore, it will be necessary in future work to examine in detail the relationships between LCC and ECTEI changes in CMIP models if we are to use ECTEI to understand low cloud feedback in models.

### d. Physical meaning of the CTE criterion and ECTEI

The criterion related to cloud-top entrainment [inequality shown in Eq. (1)], which is used as a base in the present study, was originally developed as a pure CTE instability criterion (e.g., Randall 1980). Subsequently, it has been recognized that the criterion, where *k* ≈ 0.7, is a break-up criterion of low clouds as a result of a balance among several physical processes, such as a production process by radiative cooling and dissipation processes by precipitation production and ventilation effect of boundary layer humidity due to cumulus transport (e.g., Yamaguchi and Randall 2008; Lock 2009; Wood 2012). On the other hand, it is true that the CTE mechanism still plays an important role as a dissipation term of cloud layers and the criterion corresponds to the relative importance of CTE among several competing processes. Although discussion of low cloud feedback based on ECTEI implies that the magnitude of positive low cloud feedback might be attributed to increased importance of dissipation terms (including activated CTE and the cumulus ventilation effect, relative to the production terms, in a future climate), more detailed investigations are necessary for the physical interpretation. Whatever the case, the discussion presented in this study suggests the possibility of integrating various hypotheses related to low cloud feedback based on different indices using one simple index for marine low cloud cover.

### e. Constant in the CTE criterion

The present study has used the factor *k* = 0.70 in the CTE criterion in Eq. (1), which was derived by MacVean and Mason (1990) and validated by numerical simulations undertaken by MacVean (1993). However, the original form of *k*, which is proposed as in Eq. (13) of MacVean and Mason (1990), has a temperature dependency because the formulation involves temperature derivatives of saturation specific humidity. Figure 6a shows the dependency of factor *k* on SST, under the assumption of an atmospheric profile commonly used in the present study (see appendix A). The range of *k* is 0.61–0.74 for the SST range of 275–300 K. Figure 6c shows that the magnitude of a partial derivative of ECTEI with respect to SST is smaller for a temperature-dependent *k* than for constant *k* = 0.70. The magnitude of the partial derivative of ECTEI with respect to SST for a temperature-dependent *k* is smaller by 30% at SST = 293 K than for the constant *k*, although the value of the temperature-dependent *k* is almost equal to 0.70 at the SST (Fig. 6a). Figure 6b helps us understand this characteristic. Even though the difference ECTEI − EIS is the same for two cases of *k* at SST = 293 K, the gradient of the curve of ECTEI − EIS is smaller for temperature-dependent *k* than for constant *k*. Figure 6c shows that the SST slope can be smaller than the value discussed in the present study, if the temperature-dependent *k* of MacVean and Mason (1990) is more realistic than constant *k*. It is also worth noting that the sensitivity test on *β* shown in Fig. 4d can be regarded as the sensitivity to different values of the constant *k*; *β* = 0.16 and 0.30 correspond to *k* = 0.79 and 0.61, respectively, if *C*_{qgap} is constant as 0.76. We can understand from this sensitivity examination that a smaller (larger) constant *k* gives a larger (smaller) SST slope. The value of *k* has some uncertainty and additionally the value can depend on cloud regimes. The results shown in Figs. 4d and 6 can give some information for the use of a value of *k* different from a constant value 0.70. In addition, different values of *k* can be considered conceivable given that a relatively wide range of values for *β*, corresponding to a substantial range of *k*, give high correlation between LCC and ECTEI (Fig. 2). For reference, when the temperature-dependent *k* is used (the reference pressure is set to a mandatory level of 925 hPa to calculate *k* from ERA-40 data), the correlation between LCC and ECTEI is also 0.91, which is equivalent to the case of constant *k* = 0.70 (for the temperature-dependent *k*, a derivative *d*LCC/*d*ECTEI = 3.4% K^{−1}).

Although influences from different value of *k* are discussed in this section, we also consider it necessary to discuss the influence of the liquid water term on the CTE criterion if we are to consider the criterion in more detail. The more accurate representation of the CTE criterion should include a dependence on the availability of liquid water at the top of the cloud layer (e.g., Wood 2012; MacVean and Mason 1990; Duynkerke 1993). Although this issue is not discussed in the present study, it would be also an interesting research topic in the context of low cloud feedback because the liquid water content can change in the future climate.

## 5. Summary

This paper reports the development of a new index for LCC, ECTEI, which is based on the EIS index and takes into account a CTE criterion. Analysis of shipboard cloud observation data showed that this new ECTEI is strongly correlated with LCC. We have proposed a concept based on the ECTEI that can comprehensively integrate various previous findings on indices related to LCC. We have calculated the slopes of various indices using ECTEI and compared them with those reported in previous observational and model simulation studies, and have demonstrated that the EIS and SST slopes under the present climate, studied by Qu et al. (2014), can be deduced mathematically from ECTEI. Furthermore, the slope of the vertical gradient of MSE under the present climate, shown by Kubar et al. (2011), can be also explained using the concept of ECTEI.

There is a possibility that a change in ECTEI can explain a change in LCC in a future climate if the relationship persists, and hence low cloud feedback can be interpreted using ECTEI. This study proposes a possible explanation for simulated results in CMIP models indicating that low clouds are decreased in the future climate with increased SST in spite of increases in EIS. Physical interpretations of cloud feedback that have been proposed previously, based on the vertical gradients of MSE, EIS, and SST, can be comprehensively integrated using ECTEI, at least qualitatively.

It is also possible that a discussion centered on ECTEI could provide additional insights. For example, cloud reduction in the future climate may be greater in the subtropics than in midlatitudes for the same temperature increase, because the SST slope, (∂LCC/∂SST)_{EIS}, is more negative in high-SST regions (Fig. 4). In addition, our results imply that cloud reduction is greater for low-RH_{700} models for a given SST and a given SST increase. A more detailed investigation of low cloud feedback based on ECTEI will be the subject of a future study. In addition, we plan to examine the relationships between changes in LCC and ECTEI in CMIP models, in both present-day and climate change experiments.

## Acknowledgments

The EECRA data were obtained from the Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, U.S. Department of Energy. The ERA-40 and ERA-Interim data used in this study were provided by ECMWF. EPIC data were downloaded from the EPIC Stratocumulus Integrated Dataset website. This research was partly supported by the “Program for Risk Information on Climate Change” (SOUSEI), the TOUGOU Program, and “Social Implementation Program on Climate Change Adaptation Technology” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Mark Webb was supported by the Joint UK BEIS/Defra Met Office Hadley Centre Climate Program (GA01101). We thank Dr. Chris Bretherton for insightful comments. We acknowledge Dr. Stephen Klein and Dr. Robert Wood as reviewers and one anonymous reviewer for their detailed constructive and insightful comments. The figures were edited by Rikako Matsumoto.

### APPENDIX A

#### Assumed Profile of a Lower Troposphere

This appendix describes in detail the lower-tropospheric profiles typical for stratocumulus regions that are used in the present study to examine the dependencies of stability indices on SST. First, we need to provide the temperature difference between the surface air temperature and SST. For instance, de Szoeke et al. (2010) showed that the observed surface air temperature was lower than the SST by 0.5–1 K in the southeastern Pacific during Variability of the American Monsoon Systems (VAMOS) Ocean–Cloud–Atmosphere–Land Study (VOCALS) regional experiment in 2008 (Wood et al. 2011). To obtain a rough global value, the temperature difference was calculated from the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim) data (Dee et al. 2011). The data over the oceans between 60°S and 60°N during 1979 and 2008 give an average difference from −1.0 to −1.2 K depending on the month. Therefore, it is assumed in our study that the surface air temperature is 1-K lower than the SST as a rough estimate. However, note that although for simplicity the average value over the oceans is used, the temperature difference does of course vary regionally and seasonally; some regions such as the fog regions over the midlatitude oceans in summer have positive air temperature minus SST differences, while other regions such as cold-air outbreak regions over wintertime midlatitude oceans have much larger negative temperature differences. The given surface pressure is 1013 hPa.

It is assumed that the profile has a dry adiabatic lapse rate of 9.8 K km^{−1} below the LCL and a moist adiabatic lapse rate inside the cloud layer and free atmosphere. The standard profile used in the present study has a temperature inversion of 4 K, and relative humidity of 80% at the surface and 30% at 700 hPa (sensitivities of the results to the given profile are discussed in the text). The total water content below the inversion is constant and equal to the specific humidity at the surface, and the relative humidity above the inversion is the same as at 700 hPa. The moist adiabatic lapse rate at each height is calculated using the temperature and pressure of each height, and the LCL is also determined uniquely from the given temperature and humidity profiles. To determine the inversion height, the thickness of cloud layer (the distance between the LCL and inversion height) is assumed to be 300 m. Figure A1 shows vertical profiles of temperature and specific humidity for the assumed standard atmospheric structures for SST = 280 and 300 K as examples.

### APPENDIX B

#### Formulation of Indices for LCC and Dependency of the Differences on SST

The indices LTS (Klein and Hartmann 1993), EIS (Wood and Bretherton 2006), and ECTEI are defined as follows:

where the subscript 700 refers to the pressure level of 700 hPa, the subscript surf indicates the ocean surface level, is the moist adiabatic lapse rate of potential temperature at 850 hPa, *z* is height, LCL is the lifting condensation level, and *β* = (1 − *k*)*C*_{qgap}. Therefore, the difference between LTS and EIS can be written as follows:

where can be written as follows (Wood and Bretherton 2006):

Here, *g* is the gravitational acceleration, *q*_{sat} is the saturation specific humidity, *p* is pressure, and *R*_{d} and *R*_{υ} are the gas constants for dry air and water vapor, respectively. Therefore, the difference between LTS and EIS depends mainly on the average temperatures at the surface and 700 hPa, which means that the difference is also strongly dependent on SST.

On the other hand, the difference between ECTEI and EIS is

The term (*q*_{surf} − *q*_{700}) is strongly related to *q*_{sat}(*T*_{surf}) in nature because *q*_{surf} is controlled mainly by *q*_{sat}(*T*_{surf}) and *q*_{surf} is much larger than *q*_{700} in most cases. Therefore, the difference between ECTEI and EIS depends strongly on SST.

## REFERENCES

## Footnotes

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