1. Introduction
Parameterizations of cumulus ensembles significantly influence the performance of atmospheric general circulation models (AGCMs) and have long been recognized as a core component of the models. Although two-dimensional cloud-resolving models (CRMs) embedded in AGCM grids and a global CRM began to be used recently thanks to the rapid progress of computer technology (Grabowski 2001; Khairoutdinov and Randall 2001; Satoh et al. 2008), such models are very expensive to run and require a huge amount of storages for the output. The conventional AGCMs will still be necessary over a long period of time to study long-term behaviors of atmosphere–ocean coupled systems. For such practical purposes, improvement of the cumulus parameterization will still be a central issue to obtain a better climatology and variability of the climate system.
More attention should be paid, however, to the point that the development of a cumulus parameterization is also an attempt to build a conceptual model for the purpose of understanding macroscopic behaviors of cumulus convection, as argued in Lin and Arakawa (1997a) and Arakawa (2004). Complicated models do not necessarily provide understanding of complex phenomena and simplification is inevitably required on many levels. Cumulus parameterization as a conceptual model should be recognized as such—a scientific work rather than a practical or engineering one—and will be useful to interpret results of CRMs even when global CRMs can be handled with much less effort in the far-off future.
The process of entrainment has been an important but still controversial issue for cumulus parameterization. One of the discussions has been whether an entraining plume model derived from an analogy with laboratory water-tank experiments is appropriate for cumulus updraft in the atmosphere. A number of studies have argued that entrainment can occur through the cloud top (Squires 1958; Paluch 1979) and that mixing of cloud air parcels with surrounding air occurs episodically rather than continuously (Paluch 1979; Raymond and Blyth 1986; Emanuel 1991).
On the other hand, Taylor and Baker (1991) demonstrated that observations given in Paluch (1979) that had been considered to support the idea of episodic mixing can be interpreted from a viewpoint of continuous lateral mixing. Lin and Arakawa (1997a) analyzed the origins of entrainment sources simulated in a two-dimensional CRM of 500-m horizontal resolution by an elaborate method using trajectory analysis for a deep convection case. They showed that the surrounding air enters into clouds continuously and mainly through their lateral boundary, suggesting that the entraining plume model is generally an acceptable simplification. Afterward, this view was supported by Heus et al. (2008) for a shallow convection case. Lin and Arakawa (1997b) further examined a spectral cumulus ensemble model as in the Arakawa–Schubert scheme (Arakawa and Schubert 1974) and suggested that the model is adequate if different types of clouds in the spectrum are interpreted as subcloud elements with different entrainment characteristics.
Lin and Arakawa (1997b) also found out that the fractional rate of entrainment (hereafter simply entrainment rate) is not a constant in height but tends to be larger at lower levels and near cloud top. Similar vertical profiles were obtained by several studies (Lin 1999b; Cohen 2000; Swann 2001; Murata and Ueno 2005) for different cases, models, resolutions, and methods of analysis. Estimations of the entrainment rate for trade cumuli from large-scale budgets of heat and moisture observed during the Barbados Oceanographic and Meteorological Experiment also showed larger values near cloud base (Esbensen 1978) and smaller ones near cloud top, unlike deep convection. A similar profile was obtained by an aircraft observation of trade cumuli off the coast of Hawaii (Raga et al. 1990). A large number of trade cumulus experiments with large-eddy simulations (LESs) reproduced the same profile (Siebesma and Cuijpers 1995; Grant and Brown 1999; Siebesma et al. 2003). Stevens et al. (2001) showed that the rate is larger near cloud top even for the trade cumulus simulated by LESs under the boundary conditions of the Atlantic Trade Wind Experiment where stratocumulus is accompanied at the top of the cloud.
While there are a number of arguments that the vertical variation of entrainment rate can be estimated simply by a function of height (Siebesma 1998; Swann 2001), several other works showed that it varies from case to case even if the cloud tops lie at a similar level, suggesting that the profile is not dependent on height alone (Lin 1999b; Cohen 2000). Lin (1999b) demonstrated that the simulated profiles are sensitive to thermodynamic fields but not to vertical wind shear. Based on a statistical analysis of a cumulus ensemble, Lin (1999b) proposed an empirical relation that entrainment rate at any level of clouds is proportional to the parcel buoyancy to the power of 
The entrainment rate is commonly considered to be linked with cloud radius (Simpson and Wiggert 1969); however, its complex variabilities, mentioned above, have motivated new approaches. Grant and Brown (1999) made a similarity argument about the entrainment rate analogous to those applied to the boundary layer. They assumed that a certain fraction of kinetic energy generated by parcel buoyancy is reduced by entrainment processes. With an LES, they demonstrated that different values of entrainment rate for shallow cumuli with differing depths can be explained by the assumption.
While the argument in Grant and Brown (1999) was intended to be applied for the vertical mean entrainment rate rather than vertical profiles, Gregory (2001, hereafter GR01) adopted this assumption to estimate vertical profiles of entrainment rate for both deep and shallow cumuli by modifying the formulation into a more general but relatively crude form. The new formulation was implemented in the Gregory scheme (Gregory et al. 2000) and a single-column experiment with the scheme showed large entrainment rate near cloud base.
Neggers et al. (2002, hereafter NSJ02) developed a shallow convection scheme that considers the coexistence of multiple air parcels in individual shallow cumuli and showed that the scheme reproduces the probability distribution of in-cloud properties simulated in an LES. In their formulation, entrainment was simply considered to be a process to relax in-cloud properties into that of the surrounding environment with a certain time scale. With this assumption, the entrainment rate is inversely proportional to updraft velocity. Cloud air parcels with different in-cloud properties and upward velocity at cloud base have different vertical profiles of entrainment rate.
In the latest version of the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System (Cy32r3), the entrainment rate of the cumulus scheme was formulated so that it varies vertically and depends on environmental humidity (Bechtold et al. 2008). The new formulation is not physically based; however, it produced excellent results in both climatology and variability including a large signal of the Madden–Julian oscillation (MJO), suggesting that entrainment rate is a critical factor controlling model performances.
This study develops a cumulus parameterization where entrainment rate vertically varies depending on surrounding environment and implements it in an AGCM. Two different formulations proposed by GR01 and NSJ02 are examined for the rate. Although the formulations are still controversial, they are at least based on physical considerations. Following the spirit of the Arakawa–Schubert scheme, multiple cloud types are considered and spectrally represented. In section 2, the formulation of the new scheme is described. Instantaneous responses of the scheme to different surrounding environmental profiles in a single-column model are examined in section 3. The designs of the AGCM experiments and the results are shown in section 4. This paper focuses mainly on sensitivities of the scheme to different temperature and humidity profiles. The climatology of the AGCM result is fully discussed in a companion paper (Chikira 2010). Finally, a summary and discussion are presented in section 5.
2. Description
The governing equations describing the interaction between a cumulus ensemble and the large-scale environment follow Arakawa and Schubert (1974) where tendencies of energy and water in the environment are caused by detrainment and compensating subsidence due to upward cumulus mass flux. An effect of cumulus friction is included according to Gregory et al. (1997). As discussed in Lin and Arakawa (1997a), the entraining-plume model is considered to be a good approximation of cumulus updraft and is adopted here.
a. Entrainment rate
This formula captures the fact that more energy is required to accelerate entrained air to a larger updraft velocity and the energy ultimately derives from the buoyancy. A possible interpretation might be that if buoyancy production is not sufficient, the surrounding air that was initially entrained into clouds cannot reach the speed of rising cloud parcels and thereby finally detaches from the cloud without homogenizing. After several tests, Cϵ is set at 0.6, which is close to the value used in GR01 for a shallow cloud type.
b. In-cloud properties
c. Spectral representation
Following the spirit of the Arakawa–Schubert scheme, cloud types are spectrally represented. The Arakawa–Schubert scheme considered different types of clouds according to different values of the entrainment rate. In the scheme developed here, however, the entrainment rate is calculated using either Eq. (2) or (3).
One possible method is to consider that the tuning parameters such as Cϵ in Eq. (2) and μ in Eq. (3) are distributed in a certain range. Lin (1999b) adopted this direction. In that work, the entrainment rate was expressed as proportional to the parcel buoyancy to the power of
This study takes a different approach. As discussed in NSJ02, one interesting characteristic of the framework described in the previous subsections is that vertical profiles of in-cloud properties, as well as cloud-top height, are influenced by properties at cloud base. For instance, with Eq. (3) a cloud parcel with larger ŵ, ĥ, and q̂t at the cloud base has initially a smaller entrainment rate and larger buoyancy, which leads to less entrainment and more acceleration of the parcel. This process works as a chain of a positive feedback and finally the parcel reaches a higher level. With smaller ŵ, ĥ, and q̂t, a cloud parcel tends to have larger entrainment rate and less buoyancy, leading to a lower cloud top. In case of Eq. (2), the feedback is more complicated since the entrainment rate also depends on buoyancy; however, a similar feedback occurs near cloud base as shown in section 3.
NSJ02 developed a diagnostic model of shallow convection. They input cloud-base properties of many rising parcels simulated in an LES experiment of shallow convection to the scheme. Then they succeeded in reproducing the probability distribution of the in-cloud properties of the convection. On the other hand, this study uses this cloud-base-controlled feature to express all types of cumulus clouds including both deep and shallow convection. Obviously, updraft velocity at cloud base has a wide range of values because of the different strengths of thermals, gravity waves, and lifting induced by downdraft and orography. Also, moist static energy and total water of rising parcels are considered to be distributed in certain ranges in boundary layer. Here, spectral representation is built according to cloud-base properties. Note that the cloud type defined here does not represent an individual cloud. As discussed in Lin and Arakawa (1997b), it should be interpreted as a subcloud element.
NSJ02 used cloud-base values of liquid water temperature, total water, and updraft velocity for their model; however, the scheme proposed here is intended to be used in GCMs. It will be too burdensome to consider all three cloud-base properties and build a three-dimensional spectrum. The analysis in Lin and Arakawa (1997b) seems to show that the cloud-base values of moist static energy do not quite depend on cloud types (see Fig. 7 in Lin and Arakawa 1997b). As a first step, cloud types are represented according to cloud-base updraft velocity alone. While the minimum and maximum values of the updraft velocity are influenced by many factors, it is simply assumed that the velocity ranges between two fixed values. Furthermore, Cϵ and μ never change depending on cloud type. Verification of all these simplifications will be a subject of future studies.
In the Arakawa–Schubert scheme, a cloud-top level is first given and then the entrainment rate corresponding to the level is inversely solved. Since the formulation of entrainment rate is more complicated here, mathematically there is no guarantee that cloud-base updraft velocity that corresponds to the midst of a given cloud-top level always exists. Therefore, different values of cloud-base updraft velocity are first given from the minimum to the maximum with a fixed interval. The minimum and maximum values are set at 0.1 and 1.4 m s−1, with an interval of 0.1 m s−1, for GR01’s formula, and at 0.2 and 2.8 m s−1, with an interval of 0.2 m s−1, for NSJ02’s formula.
In-cloud properties are then integrated upward with Eqs. (2), (4), (5), (6), and (7) for the formula of GR01 and Eqs. (1), (3), (5), (6), and (7) for the formula of NSJ02. This upward integration continues even if the buoyancy is negative as long as the updraft velocity is positive. If the velocity becomes negative at some point, the parcel detrains at the neutral buoyancy level that is below and closest to the point. That is, the scheme automatically judges whether the rising parcel can penetrate the negative buoyancy layers when there is a positive buoyancy layer above. The effect of CIN near cloud base is also represented by this. Note, however, that an effect of overshooting above cloud top is not included for simplicity (i.e., detrainment never occurs above cloud top).
A numerical scheme for solving the set of the equations is devised considering accuracy, stability, and column conservation of mass, energy, and water. The details are described in appendix A. For determination of ĥ and q̂t at cloud base, see appendix B.
d. Cloud-base mass flux
Other miscellaneous treatments such as downdraft and cloudiness are described in appendix B.
3. Instantaneous responses in a single column
First, the instantaneous responses of the new schemes to different temperature and humidity profiles without interaction with the surrounding environment are examined. Unlike interactive experiments, this method allows us to isolate the effects of different factors such as CAPE, CIN, and humidity and provides a basis for understanding the AGCM results.
a. Experimental design
A profile of large-scale environment is created from the version 2.1 of the Intensive Flux Array averaged fields of the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (Ciesielski et al. 2003) by taking an average of profiles where rainfall rate is more than 20 mm day−1 for the duration of three cruises (10 November–10 December, 18 December–23 January, and 31 January–18 February). It is known that ĥ and q̂ at cloud base have larger values than
All the tuning parameters were adjusted trying to obtain the best performance of climatology and variability in the AGCM; however, as one can anticipate, these settings do not provide good results for single-column tests of this kind. The values of Cϵ in Eq. (2) and μ/τc in Eq. (3) are modified to 0.5 and 1.8, respectively, only in these single-column tests. The changes are small and do not influence the qualitative discussions of the schemes.
b. Response of each cloud type
Figure 1 shows the responses of all the cloud types to the surrounding environment in each scheme. An outstanding feature in GRE and NSJ is that the entrainment rate is notably large near the cloud base, consistent with many CRM results. The feature is explained by the fact that the updraft velocities are small near the cloud base in both GRE and NSJ. As a result, ĥ is gradually reduced upward in all the cloud types. The entrainment rate tends to be slightly enlarged near the cloud top in NSJ because of the reduction in updraft velocity. The stronger upward motion at the cloud base leads to the larger entrainment rate and smaller ĥ. The cloud top tends to be higher with the larger cloud-base updraft velocity especially in NSJ.
In GRE, the entrainment rate for each cloud type differs only in the lower troposphere from the surface up to 800 hPa and tends to converge upward. The reason seems to come from the property of Eq. (4) that an increase in ŵ2 per a unit height is in proportion to B, which makes the ratio B:ŵ2, and thereby the entrainment rate defined by Eq. (2), converge to a fixed value upward. This point seems to explain why the cloud tops in GRE do not tend to differ according to cloud type, in contrast to NSJ and PAS.
A large difference is seen in the magnitudes of the updraft velocities between GRE and NSJ. The NSJ results of updraft velocity and buoyancy seem to agree better with the CRM results (Xu and Randall 2001). While it is easy to change the result of GRE by tuning only for these single-column tests, such changes make the AGCM performance worse.
c. Sensitivity to CAPE
Figure 2 shows the results where the surrounding environment is the same but with different cloud-base height. Here, ĥ and q̂ at cloud base are given by
The entrainment rates in GRE do not show large differences. The reduction in buoyancy leads to the reduction in the updraft velocity. Since the effects of the buoyancy and updraft velocity tend to cancel out, the entrainment rates tend to be unchanged. The cloud top does not seem to largely depend on CAPE. On the other hand, the cloud top in NSJ is sensitive to the reduction of CAPE. The rates are larger with the smaller CAPE because of the rapid reduction in the updraft velocity, which effectively leads to suppression of deep convection.
d. Sensitivity to humidity
Sensitivity to different humidity profiles is shown in Fig. 3. Vertically fixed values of relative humidity are applied to the specific humidity of the large-scale environment above the cloud base from 0.1 to 0.8 with an interval of 0.1 while keeping temperature unchanged. The results show that near cloud base the buoyancy is larger with the dryer environment despite ĥ being smaller than
Despite the smaller entrainment rate above 900 hPa in GRE and below 800 hPa in NSJ for the dryer environment, ĥ is more reduced upward through the dryer environment. It means that an effect of dilution through the mixing with the dry environmental air is stronger than or comparable to that of reduction in the entrainment rate in lowering the in-cloud moist static energy. Consequently, cloud top tends to be lowered when the environment is dry. In the case of PAS, the in-cloud moist static energy is barely influenced by the environmental air. The difference occurs because the entrainment rate of the highest cloud type in PAS is much smaller than in GRE and NSJ, especially near the cloud base.
The sensitivity of the two schemes to humidity is consistent with many observational studies (Numaguti et al. 1995; Brown and Zhang 1997; Sherwood 1999; Sherwood and Wahrlich 1999; Bretherton et al. 2004b; Sobel et al. 2004; Biasutti et al. 2006; Peters and Neelin 2006; Takayabu et al. 2006; Neelin et al. 2008) and CRM results (Tompkins 2001; Redelsperger et al. 2002; Grabowski 2003; Derbyshire et al. 2004), which showed suppression of deep convection under dry condition. These results suggest that the effect of dry environment on convection is explained by vertical variation of the entrainment rate.
e. Sensitivity to CIN
An experiment is made in which the temperature of the two grids overlying the cloud base is increased by 1.3 K to create a negative buoyancy layer to examine the impact of CIN (Fig. 4). The entrainment rate in GRE is small near 950 hPa because of the small buoyancy and then is slightly increased by somewhat enlarged buoyancy and decelerated updraft velocity near 900 hPa. The in-cloud moist static energy profiles show that the effects of these changes amost cancel out. On the other hand, the rate in NSJ shows systematically large values because of the significantly reduced updraft velocity. Then the cloud top becomes much lower. The result of PAS is barely influenced by CIN. Traditionally it has been considered that CIN tends to suppress convection through deceleration of rising plumes by negative buoyancy layer. The results here show that the effect of CIN may be strengthened more by changes in the entrainment rate.
4. AGCM experiments
a. Experimental design
The AGCM used in this study is the atmospheric part of version 4.1 of the Model for Interdisciplinary Research on Climate (MIROC4.1; Hasumi and Emori 2004). The model is a developing version and is characterized by three-dimensional primitive equations in a hybrid σ–p coordinate with spectral and semi-Lagrangian hybrid discretization, a PDF-based prognostic cloud scheme (Watanabe et al. 2008), a two-stream k-distribution scheme for radiation with 111 channels (Sekiguchi and Nakajima 2008), level 2.5 of the Mellor–Yamada turbulence scheme (Mellor and Yamada 1982) revised by Nakanishi and Niino (2004), an orographic gravity wave drag, the land surface model Minimal Advanced Treatments of Surface Interaction and Runoff (MATSIRO), and prognostic aerosols with direct and indirect effects (Takemura et al. 2005).
The resolution is T42 with 56 levels. A relatively higher vertical resolution is adopted in light of the importance of the vertical variation in entrainment rate. Climatological values of observation are used for sea surface temperatures and sea ice distribution. The integration is continued for 15 yr, and the last 10 yr are used for the analysis.
b. Precipitation
To provide a simple framework to understand the complex responses of the schemes to humidity profiles, the Vertically Integrated Relative Humidity index (VIRH), defined by Eq. (10), is introduced, but pb and pt in the formula are fixed at 850 and 300 hPa, respectively. The variables are then binned in a two-dimensional map of VIRH and CAPE with intervals of 5% and 500 m2 s−2, respectively, over the sea between 10°S and 10°N. Because of a limited amount of storage, all of the binning processes in this work are based on the daily outputs of 10 yr.
Figure 5 shows the bin mean cumulus precipitation and the ratio of its standard deviation to the mean. The distribution shows that AS is to a large extent exclusively sensitive to CAPE (Fig. 5a). The feature is well understood from the formulation of cloud-base mass flux in Eq. (8), in which the flux is increased by buoyancy. It is not sensitive to humidity, consistent with the single-column test. On the other hand, the inclusion of the cumulus suppression scheme in AS brings a drastic change. Figure 5b shows that VIRH becomes a major controlling factor, as anticipated.
GRE and NSJ are sensitive to both CAPE and VIRH. The sensitivity to VIRH seems to be consistent with the single-column tests. Although GRE in the single column is not very sensitive to CAPE, the AGCM result shows a large dependency, suggesting that the tests for only one surrounding profile cannot cover the whole situation that occurs in the tropics.
In GRE and NSJ, CAPE tends to be larger than observational values of approximately 2000–3000 m−2 s−2 at most. The reduction in ĥ with height owing to a larger entrainment rate near cloud base in GRE and NSJ tends to keep larger CAPE values, since the definition of CAPE is vertically integrated buoyancy of surface air without entrainment. One of the causes of this overestimation presumably comes from the method for determining cloud-base properties. Although ĥ should be larger than
c. Cumulus heating
Figure 6 shows the bin mean cumulus heating. In Figs. 6a–d, CAPE is sequentially changed with the same interval while keeping VIRH unchanged. As seen in the previous section, AS, GRE, and NSJ are sensitive to CAPE, while ASRH is not; however, the heating profiles are quite different among AS, GRE, and NSJ. One feature in GRE and especially NSJ is that the heating tends to occur in low levels when CAPE is small, while in AS it tends to occur at high levels. Also, the heating in GRE generally tends to be larger in the middle troposphere than in the other schemes.
In Figs. 6e–h, VIRH is sequentially changed with the same interval while keeping CAPE unchanged. In this case, ASRH, GRE, and NSJ are sensitive to VIRH, while AS is not; however, features of the dependency are different among ASRH, GRE, and NSJ. The heating profile in ASRH is quickly reduced and shifts to the lower troposphere near 70% in VIRH, while in GRE and NSJ it changes smoothly along with VIRH in both magnitude and height. The difference of the behavior of ASRH from that of GRE and NSJ can be understood by the formulation of the suppression scheme, which only triggers or suppresses convection based on a threshold of humidity. Again, the heating profile in GRE generally tends to be larger in the middle troposphere than the other schemes.
d. In-cloud properties
Hereafter, analyses concern only GRE and NSJ for brevity. Figures 7a–d show the bin mean in-cloud moist static energy, entrainment rate, buoyancy, and updraft velocity in GRE. The profiles are shown as to the cloud type with the maximum updraft velocity at its cloud base. CAPE is sequentially changed while keeping VIRH unchanged.
Figure 7a shows that the difference of ĥ and
The buoyancy profiles show three peaks near 850, 650, and 350 hPa (Fig. 7c). These features correspond well to three major regimes of cumulus convection—trade cumulus, cumulus congestus, and cumulonimbus—which are often seen and discussed in observations (Johnson et al. 1999). An examination of vertical profiles of the bin frequency shows that the three peaks are not artificially created by overlapping of different clouds with different cloud tops (not shown). The feature should be explained by the mean thermodynamic profiles. The updraft velocity profiles correspond well to the buoyancy and the peaks are slightly shifted upward, as is easily predicted from Eq. (4).
Three peaks are also seen in the entrainment rate near the same levels (Fig. 7d). Considering Eq. (2), the feature is interpreted as an effect of the large buoyancy at these levels; however, the values of the peaks gradually decrease upward with the increase in the updraft velocity. The large values of the rate near 650 hPa are considered to help cloud parcels detrain near 600 hPa and generate more cumulus congestus. Chikira (2010) shows that the annual mean detrainment in the middle troposphere greatly increases in GRE. This feature explains why the cumulus heating of GRE tends to be larger in the middle troposphere compared to the other experiments. Unlike the single-column test, the entrainment rate largely depends on CAPE. Below 750 hPa, it tends to increase for larger CAPE, presumably because of the larger buoyancy. On the other hand, above 750 hPa it tends to be reduced for larger CAPE, presumably because of larger updraft velocity.
A primary factor behind the scheme’s sensitivity to CAPE seems to be the reduction in ĥ owing to the larger entrainment rate, which tends to suppress deep convection when CAPE is small. The modulation of entrainment rate might influence on the sensitivity; however, the direction of the change in the rate against CAPE is opposite below and above 750 hPa. It is not clear how this feature modifies the scheme’s sensitivity to CAPE.
Figures 7e–h are the same as Figs. 7a–d, but VIRH is sequentially changed while keeping CAPE fixed. The general features of the entrainment rate, buoyancy, and updraft velocity are similar to Figs. 7a–d. Again the result is different from the single-column test. The profiles of ĥ and
Considering the result of the single-column test, a primary factor behind the scheme’s sensitivity to VIRH seems to be the large values of the entrainment rate near the cloud base. As a secondary factor, the modulation of the rate seems to strengthen the sensitivity.
In Figs. 7b and 7f the causes of the differences from the single-column tests are not sufficiently clear, although it should be associated with the changes in the profiles of the surrounding environment. The effects of the buoyancy and updraft velocity on the entrainment rate tend to cancel out and the rate seems to be determined by the subtle difference of the two variables.
Figures 8a–d are the same as Figs. 7a–d, but for NSJ. The differences of ĥ and
Figures 8e–h are the same as Figs. 7e–h, but for NSJ. Unlike the single-column test, the entrainment rate is larger when the surrounding environment is drier. As in GRE, CIN tends to be larger in a drier environment (Fig. 8e), which thereby reduces the updraft velocity near cloud base (Fig. 8h), amplifying the entrainment rate (Fig. 8f). As suggested by the single-column test, the moisture effect on the virtual potential temperature of the cloud parcel should enlarge the buoyancy more in the drier environment, leading to a reduced entrainment rate; however, this effect does not overcome the CIN effect.
e. Examples of climatology and variability
For the reader’s convenience, examples of the effects of the new schemes on the climatology and variability are shown. The climatology is discussed in the companion paper in detail (Chikira 2010). An examination of the variability is ongoing and planned to be published in the future.
The annual mean precipitation is shown in Fig. 9. Note that the tuning parameters of the AGCM are optimized for GRE and NSJ. It is meaningless to discuss the subtle differences. The result of ASRH is qualitatively similar to an old version of the MIROC, although the version is quantitatively better because of the best adjustment of the parameters.
First, in AS and ASRH, precipitation is underestimated over the water and relatively more precipitation tends to be produced over land, especially the Maritime Continent. In contrast, GRE and NSJ show better agreement with observations. Second, the South Pacific convergence zone (SPCZ) is better represented in GRE and NSJ. Third, AS and ASRH show a trace of a double intertropical convergence zone (ITCZ) in the eastern Pacific, while in GRE and NSJ the southern side of the ITCZ is weakened, consistent with the observation.
Figure 10 shows the zonal wavenumber–frequency power spectra of the symmetric and antisymmetric components of the outgoing longwave radiation divided by the background power (the so-called Wheeler–Kiladis diagram), which was preprocessed according to a series of procedures described in Wheeler and Kiladis (1999), using the results of all the experiments (10 yr) and the Advanced Very High Resolution Radiometer (AVHRR; from 1979 to 2005; Liebmann and Smith 1996) between 15°S and 15°N including all the seasons.
Correspondence of the signals with dispersion curves shows that n = 1 equatorial Rossby (ER) waves are well represented in all the experiments [hereafter, n denotes a meridional mode number as in Matsuno (1966)]. Equatorial Kelvin waves are also seen in all the experiments; however, in AS and ASRH the equivalent depths are overestimated, as is seen in most GCMs (Lin et al. 2006). On the other hand, in GRE and NSJ the depths are consistent with the observations. As for mixed Rossby–gravity (MRG) and n = 0 eastward inertio-gravity (EIG) waves, only ASRH and NSJ show signals of reasonable strength. The equivalent depth in NSJ agrees well with the observation, while that in ASRH does not. Note that in Fig. 10 “WIG” denotes an n = 1 westward inertio-gravity wave. As for the MJO, the corresponding signal is the strongest in GRE and comparable to the observation.
5. Summary and discussion
A new cumulus parameterization was developed in which the entrainment rate varies vertically depending on surrounding environment. An entraining plume model was adopted. Two formulations by GR01 and NSJ02 were examined. Cloud types of different cloud height are spectrally represented according to different values of updraft velocity at cloud base. This framework provides a new approach to understanding why clouds of different cloud-top height coexist in the same large-scale environment, which Arakawa and Schubert (1974) tried to explain in terms of different values of entrainment rate for different cloud types. Cloud-base mass flux is determined with prognostic convective kinetic energy closure.
The new schemes were compared with a variant of the Arakawa–Schubert scheme with and without a cumulus suppression scheme based on environmental humidity. Both formulations produced a large entrainment rate near cloud base because of the small updraft velocity, leading to a gradual reduction upward in the moist static energy of a rising cloud parcel. The rate tends to be slightly increased near cloud top in NSJ as a result of decelerated updraft velocity. These features are qualitatively consistent with many of the previous CRM results. The weaker upward motion of cloud air at cloud base leads to larger entrainment rate and smaller in-cloud moist static energy, thereby leading to a lower cloud top.
The AGCM results show that cumulus activity of AS mainly depends on CAPE, while that of ASRH is exclusively controlled by VIRH. On the other hand, GRE and NSJ are sensitive to both CAPE and VIRH. In both GRE and NSJ, cloud top tends to be gradually lowered when CAPE approaches small values, while AS and ASRH show little sensitivity in cloud-top height. There are two evident factors for it. One is that the vertical profile of environmental saturation moist static energy tends to approach a vertical line. In this situation, deep convection tends not to occur because of a reduction in the moist static energy of the rising cloud parcel caused by a larger entrainment rate near cloud base. The other factor is the change in entrainment rate. In NSJ, the updraft velocity tends to be weakened by small CAPE, thereby increasing the entrainment rate. In case of GRE the change of the rate is complicated and its effect on the sensitivity is not very clear.
A primary factor behind the scheme’s sensitivity to VIRH seems to be large values of entrainment rate near cloud base in both GRE and NSJ, which rapidly reduces in-cloud moist static energy, leading to lower cloud top. As a secondary factor, modulations in the entrainment rate tend to strengthen the sensitivity in the both experiments. It seems that there are two major factors for enlarging entrainment rate. One is reduction in updraft velocity caused by enhanced dilution and loss of buoyancy. The other is a CIN effect. CIN is negatively correlated with environmental humidity and it tends to reduce updraft velocity through a negative buoyancy layer near cloud base, leading to a larger entrainment rate. In addition, when environmental air is very dry, an effect of in-cloud moisture on its virtual potential temperature greatly enlarges buoyancy and then updraft velocity, which leads to a smaller entrainment rate. These effects, in total, slightly increase the entrainment rate in both GRE and NSJ. The sensitivity of the two schemes to humidity is consistent with many observational studies and CRM results, suggesting that the effect of dry environment on convection is explained by vertical variation of the entrainment rate. Unlike ASRH, which shows a quick transition between low and high cloud, GRE and NSJ show that cloud top gradually varies along with VIRH, generating more cumulus congestus.
In GRE and NSJ, the AGCM results are greatly improved in both climatology and variability. The annual mean precipitation is improved in many points, such as the continental precipitation, SPCZ, and double ITCZ. The representation of equatorial waves is improved without using empirical triggering schemes. In particular, GRE exhibits a signal of the MJO.
A major limitation of this study is that verifications of the theories on entrainment rate are not sufficient yet. Although many CRM and LES studies reveal that the entrainment rate varies vertically, sensitivities of the rate to thermodynamic and humidity fields have not been investigated sufficiently and systematically. Traditionally it has been considered that the entrainment rate is linked to cloud radius; however, the radius never appears in the two formulas of the rate adopted in this work. At least the formulas do not contradict the traditional understanding, since larger cloud size tends to be accompanied by larger updraft velocity, which leads to smaller entrainment rate. It is still an open question as to what really controls the rate.
In GRE, the difference of in-cloud moist static energy and environmental saturation moist static energy seems to be too small, leading to too small buoyancy and updraft velocity; however, it is premature to conclude that the formulation of entrainment rate in GRE is not adequate. As a physical theory, this formulation is more attractive than that in NSJ in that it considers redistribution of buoyancy-generated energy. In addition, GRE is better than NSJ in climatological precipitation and the representation of the MJO-like wave. Verifications with process-resolving models and reexamination of the formulation may lead to better results in the future.
Another limitation of this work is that the spectral representation is built according to cloud base updraft velocity alone. Its verification is not sufficient and fluctuations in moist static energy and moisture at cloud base might largely modify vertical structure of clouds through the buoyancy effect on the entrainment rate near cloud base.
Furthermore, the minimum and maximum values of the cloud-base updraft velocity are fixed in this work. Obviously it should vary depending on circumstances of boundary layer, downdraft activity of already existing deep convection, orographically induced gravity waves, and so on. The framework proposed by this work implies that the maximum value of the updraft velocity is reduced without downdraft from already existing deep convection, which leads to increased population of shallow convection. A nocturnal inversion near land surface and weak plumes are considered to suppress deep convection. In contrast, daytime heating over land induces a strong rising motion of air parcels in the convective boundary layer, which is favorable for the formation of deep convection with a small entrainment rate. Thus, convective activity could be more tightly linked with the boundary layer. These processes seem to be important for diurnal variation of convection. All of these bring more complexities to the feedbacks through the entrainment rate and need extensive studies in the future.
Despite these limitations, this study reveals that variations in the entrainment rate have a potential to significantly influence convection through many feedbacks. An interesting future study will be on the role of entrainment rate in mesoscale organization of convection. Moreover, the new scheme developed in this work solves many problems of AGCMs in climatology and variability that have been limiting the performance of the models so long. Benefits obtained by development of these schemes are not limited to better performances in AGCMs. Results of CRMs and LESs are so complicated that it is difficult to understand the behaviors without referring to results of simplified models. As one of such conceptual models, this new scheme seems to provide a new insight into and better understanding of the interactions between cumuli and surrounding environment and atmospheric dynamics in the tropics, both in climatology and variability.
Acknowledgments
We would like to appreciate Professor Akio Arakawa for many thoughtful comments. Thanks are also extended to Professor Taro Matsuno, Professor David A. Randall, Dr. Tatsushi Tokioka, Associate Professor Masaki Satoh, and Dr. Kuan-Man Xu. I also appreciate the anonymous reviewers for refining this paper. The computations were performed with the Earth Simulator in JAMSTEC. The last figure was made with a diagnostic tool developed by the U.S. CLIVAR MJO Working Group.
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APPENDIX A
Numerical Procedure for In-Cloud Properties
The procedure for computing in-cloud properties is analogous to Bretherton et al. (2004a). While they did not discuss aspects of accuracy, stability, and conservation of mass, energy, and water, their method has good features on these points, as clarified below.
Normalized mass flux
In-cloud properties
Updraft velocity
Iteration
The remaining unknowns are ϵk, Bk, and ŵk. These quantities are obtained by an iterative method. At the first step, ϵk−1/2, Bk−1/2, and ŵk−1/2 are used for ϵk, Bk, and ŵk, respectively. Then quantities at k + ½ are obtained through the set of the equations above. At the second step, ϵk, Bk, and ŵk are calculated as the mean values of the variables at the adjacent half levels. Then all the quantities at k + ½ are recalculated with them.
APPENDIX B
Miscellaneous Descriptions
The other miscellaneous treatments in the scheme are identical with a variant of the prognostic Arakawa–Schubert scheme that had been used in MIROC. The original description was written in Numaguti et al. (1997); however, the details were revised. Here, up-to-date descriptions of the issues are briefly presented.
Microphysics
Melting and freezing of precipitation occurs depending on wet-bulb temperature of large-scale environment and cumulus mass flux.
Cloud-base properties
Cloud base is determined as a lifting condensation level of the lowest model level. The normalized mass flux below the cloud base is given by η = (z/zb)1/2. Then, in-cloud properties at the cloud base are solved by Eqs. (A1)–(A4).
Evaporation and downdraft
Cloudiness
In-cloud moist static energy (105 J kg−1), entrainment rate (km−1), buoyancy (10−2 m s−2), and updraft velocity (m s−1) of all cloud types in (a)–(d) GRE, (e)–(h) NSJ, and (i),(j) PAS (color lines). The same cloud type is expressed with the same color. The profiles of the in-cloud properties are shown from cloud base to cloud top for each cloud type. Moist static energy and saturation moist static energy of the large-scale environment are indicated in (a),(e), and (i) by black dashed and solid lines, respectively. The ordinate is the hybrid σ–p coordinate defined in Arakawa and Konor (1996). The value of η is close to p/ps in the lower troposphere and approaches p upward, where p and ps denotes pressure and surface pressure, respectively.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
In-cloud moist static energy (105 J kg−1), entrainment rate (km−1), buoyancy (10−2 m s−2), and updraft velocity (m s−1) of the cloud type that has a maximum updraft velocity at cloud base, with different cloud-base properties, in (a)–(d) GRE and (e)–(h) NSJ (color lines). A result with the same cloud-base property is expressed with the same color. The profiles of the in-cloud properties are shown from cloud base to cloud top. Moist static energy and saturation moist static energy of large-scale environment are indicated in (a) and (e) by black dashed and solid lines respectively. (i) As in (a), but for PAS and shows the cloud type with the minimum entrainment rate. The ordinate is the hybrid σ–p coordinate.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
In-cloud moist static energy (105 J kg−1), entrainment rate (km−1), buoyancy (10−2 m s−2), and updraft velocity (m s−1) of the cloud type that has a maximum updraft velocity at cloud base in response to different humidity profiles in (a)–(d) GRE and (e)–(h) NSJ (solid color lines). Moist static energy of the large-scale environment is indicated by dashed color lines in (a) and (e). A result with the same humidity profile is expressed with the same color. The profiles of the in-cloud properties are shown from cloud base to cloud top. Saturation moist static energy of large-scale environment are indicated in (a) and (e) by black solid lines. (i) As in (a), but for PAS and shows the cloud type with the minimum entrainment rate. The ordinate is the hybrid σ–p coordinate.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
In-cloud moist static energy (105 J kg−1), entrainment rate (km−1), buoyancy (10−2 m s−2), and updraft velocity (m s−1) of the cloud type which has a maximum updraft velocity at cloud base in response to a reference profile (solid) and modified profiles (dashed) in (a)–(d) GRE and (e)–(h) NSJ. Moist static energy and saturation moist static energy of the large-scale environment is indicated by blue and red lines, respectively, in (a) and (e). The profiles of the in-cloud properties are shown from cloud base to cloud top. (i) As in (a), but for PAS and shows the cloud type with the minimum entrainment rate. Ordinate is the hybrid σ–p coordinate.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
(a)–(d) Bin mean cumulus-induced precipitation (shading; mm day−1 with log10 scale) and (e)–(h) ratio of its standard deviation to the mean (shading) in each experiment. Frequency (yr−1 with log10 scale) is indicated by contours. Values where the frequency is less than 10 yr−1 are masked. See text for details.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
Bin mean cumulus heating (K day−1) in each experiment. In (a)–(d), CAPE (m−2 s−2) is sequentially changed with the same interval while keeping VIRH (%) unchanged. In (e)–(h), VIRH is sequentially changed with the same interval while keeping CAPE unchanged. The values of CAPE and VIRH are indicated in each figure. Note that the GRE abscissa scales are different than the other experiments.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
Bin mean (a) in-cloud moist static energy (105 J kg−1; solid lines), (b) entrainment rate (km−1), (c) buoyancy (m s−2), and (d) updraft velocity (m s−1) in GRE. All of the in-cloud variables are shown as to the cloud type with the maximum updraft velocity at its cloud base. Bin mean saturation moist static energy of the large-scale environment is shown in (a) (105 J kg−1; dashed). CAPE is sequentially changed with the same interval while keeping VIRH 87.5%. The values of CAPE (m2 s−2) are indicated in (a). The same bin is indicated by the same color. Values where the frequency is less than 10 yr−1 are masked. The ordinate is the hybrid σ–p coordinate. (e)–(h) As in (a)–(d), but VIRH (%) is sequentially changed with the same interval while keeping CAPE at 4750 m2 s−2. The values of VIRH are indicated in (e).
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
As in Fig. 7, but for NSJ.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
Annual mean precipitation (mm day−1) in (a) AS, (b) ASRH, (c) GRE, (d) NSJ, and (e) Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP) (Xie and Arkin 1997), averaged between 1979 and 1999. Contour interval is 2 mm day−1.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1
Zonal wavenumber–frequency power spectra of symmetric and antisymmetric components of outgoing longwave radiation (OLR) divided by the background power according to Wheeler and Kiladis (1999) in (a) AS, (b) ASRH, (c) GRE, (d) NSJ, and (e) AVHRR. Dispersion curves of the even and odd meridional mode-numbered equatorial waves for the three equivalent depths of 12, 25, and 50 m are indicated by red lines. See text for the abbreviated terminology of the waves. The unit of frequency is cycles per day (cpd).
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3316.1