Abstract

The authors have developed a new spectral cumulus parameterization scheme that explicitly considers an ensemble of multiple convective updrafts by interpolating in-cloud variables between two convective updrafts with large and small entrainment rates. This cumulus scheme has the advantages that the variables in entraining and detraining convective updrafts are calculated in detail layer by layer as in the Tiedtke scheme, and that a spectrum of convective updrafts with different heights due to the difference in entrainment rates is explicitly represented, as in the Arakawa–Schubert scheme. A conservative and monotonic semi-Lagrangian scheme is used for calculation of transport by convection-induced compensatory subsidence. Use of the semi-Lagrangian scheme relaxes the mass-flux limit due to the Courant–Friedrichs–Lewy (CFL) condition, and moreover ensures nonnegative natural material transport. A global atmospheric model using this cumulus scheme gives an atmospheric simulation that agrees well with the observational climatology.

1. Introduction

Cumulus parameterization schemes are used in atmospheric models at horizontal resolutions of about 5 km or coarser (e.g., Mizuta et al. 2006; Kanada et al. 2008) so that the effect of subgrid-scale cumulus clouds can be taken into consideration. Simulations by models at horizontal resolutions from 1 to 3 km have some success without cumulus schemes (e.g., Lilly 1990; Posselt et al. 2008; Satoh et al. 2008; Eito et al. 2010). However, horizontal resolution on the order of 100 m is necessary to resolve individual cumulus clouds explicitly (Yamasaki 1975; Bryan et al. 2003). Thus, cumulus parameterizations are still important even now as the resolution of atmospheric models improves.

Various cumulus parameterization schemes have been developed, such as convective-adjustment schemes (e.g., Manabe and Strickler 1964; Betts and Miller 1986), Kuo schemes (Kuo 1965, 1974), and mass-flux schemes. Mass-flux schemes are widely used because they explicitly calculate subgrid-scale convective updraft and downdraft mass fluxes, and are suitable for calculation of convective transport of material.

Traditional mass-flux schemes are classified into two main types: the Arakawa–Schubert (AS) type using the simple spectral cloud model approach and the Tiedtke type using the bulk cloud model approach. There are also other types of schemes, such as the one that represents the cloud spectrum explicitly with buoyancy sorting (e.g., Emanuel 1991; Hu 1997; Raymond and Blyth 1986). The AS-type schemes (e.g., Arakawa and Schubert 1974; Moorthi and Suarez 1992; Pan and Randall 1998; Zhang and McFarlane 1995) explicitly calculate multiple convective updrafts with different heights due to differences in entrainment rates. However, they calculate each individual convective updraft as a simple entraining plume to reduce the computational cost. They calculate the values of in-cloud variables (e.g., moist static energy and water vapor) at the cloud top of the updraft from those at the cloud bottom without explicitly calculating the values at intermediate levels. In contrast, the Tiedtke-type schemes (e.g., Tiedtke 1989; Nordeng 1994; Gregory and Rowntree 1990; Kain and Fritsch 1990; Bechtold et al. 2008) calculate only one convective updraft, but calculate it as a more elaborate entraining and detraining plume. They calculate the in-cloud variables layer by layer from the cloud bottom to the cloud top. Thus, the Tiedtke-type schemes and the AS-type schemes each have their own advantages.

One way to combine the advantages of the AS and Tiedtke types is to calculate individual multiple convective updrafts explicitly and at the same time to calculate the values of in-cloud variables layer by layer from the cloud bottom to the cloud top (e.g., Nober and Graf 2005; Chikira and Sugiyama 2010; Wagner and Graf 2010). It is a good way, but the computational cost is high because of the need to calculate explicitly about 10 or more convective updrafts layer by layer. Moreover, the greater the number of vertical levels, the greater the number of convective updrafts needed to obtain a smooth distribution of cloud-top levels.

In this study, we have developed a new cumulus scheme that has the advantages of both AS and Tiedtke types, and moreover has low computational cost. The scheme represents multiple convective updrafts with different entrainment rates by calculating only two convective updrafts with large and small entrainment rates and interpolating the in-cloud variables between the two updrafts. The new scheme has been implemented in a global atmospheric model and used for climate simulations (Yukimoto et al. 2011, 2012; Mizuta et al. 2012).

In the dynamical core of our global model, a semi-implicit semi-Lagrangian scheme is adopted to permit a longer time step than that determined from the Courant–Friedrichs–Lewy (CFL) condition. However, with a long time step, convection-induced compensatory subsidence in a mass-flux-type cumulus scheme occasionally exceeds the CFL condition, especially when a large number of vertical levels are used and the intervals of the levels are small. To overcome the CFL limit, a conservative and monotonic semi-Lagrangian scheme is used for transport by compensatory subsidence in the new cumulus scheme. The monotonic scheme also has the advantage of ensuring nonnegative natural material transport.

The remainder of this paper is organized as follows. The new cumulus parameterization scheme is described in detail in section 2 and a brief model description is given in section 3. The validity of the new scheme is examined in section 4. Climatology of the model using this scheme is compared with that of the model using the AS scheme in section 5. The variability of the models is shown in section 6. Computational costs of the new scheme and other schemes are estimated in section 7. Finally, a summary and conclusions are given in section 8.

2. Cumulus parameterization

The new cumulus parameterization scheme represents a spectrum of cumulus clouds by interpolating between two convective updrafts with large and small turbulent entrainment/detrainment rates. Figure 1 is a schematic diagram of the new scheme. As in Tiedtke (1989), organized entrainment occurs through the organized inflow associated with large-scale convergence, and turbulent entrainment and detrainment occur through the turbulent exchange of mass through cloud edges. Two Tiedtke-type convective updrafts, cumulus [a] and cumulus [b], are calculated layer by layer from the cloud bottom layer up to the highest cloud top layer. Cumulus [a] has the minimum entrainment/detrainment rate and the highest cloud top, while cumulus [b] has the maximum entrainment/detrainment rate at the cloud bottom. A continuous range of convective updrafts is assumed to be present with entrainment/detrainment rates between those of cumulus [a] and cumulus [b]. In-cloud variables in the convective updrafts, such as virtual temperature, dry static energy, water vapor, and turbulent entrainment/detrainment rate, are estimated by interpolating those of cumulus [a] and cumulus [b]. Where the virtual temperatures in convective updrafts with relatively larger entrainment/detrainment rates fall below the virtual temperature in the environment at some vertical level, the updrafts lose buoyancy and are detrained at that level as organized detrainment. Above this level, the convective updraft that has the largest entrainment/detrainment rate but still has positive buoyancy is considered as cumulus [b]. This calculation is repeated layer by layer until all convective updrafts lose buoyancy and are detrained as organized detrainment.

Fig. 1.

Schematic diagram of the new cumulus scheme. Cumulus [a] has the minimum turbulent entrainment/detrainment rate and cumulus [b] has the largest turbulent entrainment/detrainment rate at each vertical level. In-cloud variables in convective updrafts with entrainment rates between those of cumulus [a] and cumulus [b] are estimated by interpolating between those of cumulus [a] and cumulus [b].

Fig. 1.

Schematic diagram of the new cumulus scheme. Cumulus [a] has the minimum turbulent entrainment/detrainment rate and cumulus [b] has the largest turbulent entrainment/detrainment rate at each vertical level. In-cloud variables in convective updrafts with entrainment rates between those of cumulus [a] and cumulus [b] are estimated by interpolating between those of cumulus [a] and cumulus [b].

a. Convective updraft

Following Yanai et al. (1973) and Nordeng (1994), and also considering cloud ice, the steady-state equations for mass flux, dry static energy, water vapor, cloud water/ice content, and cloud ice content in the convective updraft of the cloud type , whose cloud top is at the vertical level , are as follows:

 
formula
 
formula
 
formula
 
formula
 
formula

Here is height; is updraft mass flux; and are entrainment and detrainment rates per height, respectively; () is dry static energy ( is temperature, is specific heat at constant pressure, is the gravity acceleration); is specific humidity; is cloud water/ice content; is cloud ice content; is density; is specific latent heat of sublimation of ice; is specific latent heat of evaporation of water; is specific latent heat for a water/ice mix; is condensation/deposition of water vapor in the updraft; is deposition of water vapor; is freezing of cloud water; is conversion from cloud water/ice into precipitation (rain and snow); and is conversion from cloud ice into snow. The subscript () denotes the value in the updraft, and the overbar () represents the value in the environment. The cumulus clouds are classified according to the height of the cloud top and the suffix is used as the index of a classified cloud type. The values of and are obtained from

 
formula
 
formula

where is the proportion of to , given here by an empirical function of : is 1 when is below −15°C, decreasing linearly to zero as increases from −15° to 0°C, and zero when is above 0°C. When the ratio of to is given as a function of (e.g., ), is determined from Eq. (5). When is a prognostic variable, is determined so that, for example, would not be less than .

If the upward mass fluxes of all cloud types are independently calculated layer by layer using Eqs. (1)(5), calculations are necessary, where is the number of the vertical levels in the troposphere. In the new cumulus scheme, only two convective updrafts, cumulus [a] with a small entrainment/detrainment rate and cumulus [b] with a large entrainment/detrainment rate, are explicitly calculated to reduce the computational cost, and the number of calculations is only . Comparison of roughly estimated computational time among some types of cumulus schemes is shown in section 7. In-cloud variables in convective updrafts with intermediate entrainment/detrainment rates are estimated by linear interpolation between those in cumulus [a] and cumulus [b]. Since the cumulus convection is a nonlinear phenomenon, the estimation using linear interpolation is an approximation. The validity of using linear interpolation is shown in section 4.

First, the provisional values of mass flux, entrainment, and detrainment are calculated. Second, their final values are determined by a closure assumption. Hereafter, the tilde () denotes provisional values.

Provisional entrainment and detrainment consist of organized and turbulent contributions:

 
formula
 
formula
 
formula
 
formula

where the superscripts and (, ) are for cumulus [a] and cumulus [b], respectively; the superscript org () indicates organized; and the superscript trb () turbulent. Both and are common in cumulus [a] and cumulus [b].

Turbulent entrainment and turbulent detrainment are set to be equal following Tiedtke (1989), and are given by

 
formula
 
formula

where and are entrainment/detrainment rates satisfying , and and are the enhancement factors to take into account enhanced turbulence in the lower part of the cumulus [(European Centre for Medium-Range Weather Forecasts) ECMWF 2006]. The enhancement factors vary linearly from 2 at the lowest condensation level (LCL) to 1 at 1500 m above LCL. Figure 2a shows the relationship between the vertical level and . Both and are set to be equal below the level , where is the vertical level above which lifted air becomes buoyant [level of free convection (LFC)]. At the level , they are set to be and , where and are the minimum and maximum entrainment/detrainment rates, respectively, and are given by

 
formula
 
formula

In Tiedtke (1989), the entrainment/detrainment rate for deep convection is and that for shallow convection is . Above the level , a spectrum of cumulus clouds is considered. When calculating the provisional mass flux, the probability density function (PDF) of cumulus clouds for entrainment/detrainment rates between and is assumed to be constant. Above the level , is set to the constant value , and decreases with height. In Fig. 2a, the larger , the lower the cloud top, because buoyancy is lower. The calculation of and is described in  appendix A.

Fig. 2.

Schematic diagrams of the new cumulus scheme. (a) Relations between the vertical level and turbulent entrainment/detrainment rates . Black circles show of cumulus [a] and black squares show of cumulus [b] at each vertical level. Black triangles show below . (b) Vertical distribution of the provisional upward mass flux .

Fig. 2.

Schematic diagrams of the new cumulus scheme. (a) Relations between the vertical level and turbulent entrainment/detrainment rates . Black circles show of cumulus [a] and black squares show of cumulus [b] at each vertical level. Black triangles show below . (b) Vertical distribution of the provisional upward mass flux .

From Eqs. (8) to (13),

 
formula

is satisfied. Therefore, and have the same value, denoted as , and satisfy

 
formula

The variable consists of two kinds of organized entrainment:

 
formula

The variable is the organized entrainment from the layers with high moist static energy () and is given by

 
formula

where is the maximum value of below the level of minimum saturated moist static energy and , the value of which is chosen empirically. Here, the vertical profile, not the magnitude, of the provisional value is important because the magnitude of the final value is determined using a closure assumption [see Eq. (B9)]. Deep and shallow convection occur when the maximum value of lies in the boundary layer, and midlevel convection occurs when the maximum lies in the midlevel of the troposphere. The variable is the organized entrainment associated with the grid scale convergence and given by

 
formula
 
formula
 
formula
 
formula
 
formula

where is the surface height and the unit of is . The variable is a small value for the denominator not to be zero. The variable is the lower limit of , which ensures that a small amount of organized entrainment occurs even without horizontal convergence. The variable is the upper limit of , and ensures that the organized entrainment is not unnaturally large. The values of , , , and are chosen empirically. (When , the mass flux becomes proportional to , where is surface pressure.) Here, the organized entrainment is determined from the mass convergence (e.g., Lindzen 1988; Kuell and Bott 2009), not from the moisture convergence (e.g., Tiedtke 1989).

The calculation of in-cloud variables such as , , , and by discretizing Eqs. (1)(20) is shown in  appendix A.

b. Closure assumption

The final value of a mass flux of each cloud type is determined from the provisional value by using a closure assumption.

Figure 2b shows the vertical distribution of the provisional upward mass flux . As shown in Fig. 2b, can be divided into the mass fluxes of multiple cloud types, , where denotes the vertical level of the cloud top. The proportion of the provisional mass flux of the cloud type , , at the vertical level is

 
formula

The mass flux of the cloud type , , is determined from

 
formula

The value of is calculated from the closure assumption. The closure assumption used here is based on convective available potential energy (CAPE; Fritsch and Chappell 1980; Nordeng 1994; ECMWF 2006). CAPE is given by

 
formula

where is the virtual temperature. It is assumed that an ensemble of convections has the effect of decreasing over a relaxation time , and the convection of the cloud type works to decrease over the relaxation time, where is the of the cloud type . By this assumption,

 
formula

is obtained, where is the decrease of per unit time due to the convection of the cloud type , and is the truncation wavenumber of the spectral model. The variable is made dependent on model resolution as in ECMWF (2006). In the actual calculation, not only but also convective inhibition () is considered. is obtained from

 
formula

Here, below_cloud means below the level , where is satisfied. When determining the value of in Eq. (22), we use

 
formula

instead of Eq. (24) so that the mass fluxes of convective updrafts become smaller when is large.  Appendix B shows how to calculate diagnostically from Eq. (26) and how to obtain the final values of mass flux, entrainment, and detrainment from their provisional values.

Here, is independent of cloud type , but can depend on cloud type. For example, can be larger in the convection whose cloud top is lower, which weakens shallow convections. Moreover, different closure assumption can be used dependent on the cumulus height. For example, non-CAPE-type closure assumption (e.g., Tiedtke 1989; Park and Bretherton 2009) can be used in shallow convection.

c. Convective downdraft

Only a single mass flux for the convective downdraft is calculated for simplicity. The convective downdraft is calculated using the following equations:

 
formula
 
formula
 
formula
 
formula
 
formula
 
formula

where is the downdraft mass flux; and are entrainment and detrainment rates, respectively; and , , , and are evaporation of cloud water/ice, evaporation (sublimation) of cloud ice, precipitation (rain/snow), and snow in the downdraft, respectively. The subscript () denotes the value in the downdraft, and the subscript mix () denotes the value in the equal mixture of detrainment from updrafts and environmental air. Half of the detrainment from the updrafts, (but with the upper limit of ), is supposed to be equally mixed with the environmental air cooled to the wet-bulb temperature and saturated by evaporation of precipitation. When the mixture has negative buoyancy below the level of the minimum saturated moist static energy, it becomes the organized entrainment into the downdraft . The variable has an upper limit so that does not exceed . The organized entrainment at multiple vertical levels is taken into consideration unlike in Tiedtke (1989). The turbulent entrainment and detrainment have the same value and satisfy the following equations (Tiedtke 1989):

 
formula
 
formula

The air in the downdraft is saturated by the evaporation of precipitation and cloud water. When the downdraft becomes positively buoyant at a specific level, the entire downdraft mass flux detrains at that level as organized detrainment. Otherwise, the organized detrainment from the downdraft occurs within the subcloud layer.

The method used to discretize Eqs. (27)(29) is shown in  appendix C.

d. Precipitation and convective momentum transport

The conversion from cloud water to precipitation in Eq. (4) is calculated from the Sundqvist (1978) type equation. Melting of falling snow is assumed to occur at a few vertical levels near 0°C.

The vertical transports of horizontal momentum by the convective updrafts and the convective downdraft are also calculated, where pressure gradient force related to the environmental wind shear (Wu and Yanai 1994; Gregory et al. 1997) is taken into consideration.

e. Feedback to the environment

The time evolution of the variables in the environment due to the cumulus scheme is given by the following equations:

 
formula
 
formula
 
formula
 
formula

where is melting of cloud ice along with the compensatory subsidence. Using Eqs. (2)(5) and Eqs. (28)(31), Eqs. (35)(38) can be rewritten as follows:

 
formula
 
formula
 
formula
 
formula

The method used to discretize Eq. (39) is shown in  appendix D.

f. Compensatory subsidence

The first terms on the right-hand side of Eqs. (39)(42) show downward transport accompanying compensatory subsidence. To calculate the downward transport, we adopt a conservative and monotonic one-dimensional flux-form semi-Lagrangian scheme (Lin and Rood 1996), where we use the piecewise rational method (PRM; Xiao and Peng 2004) for the vertical monotonic interpolation of variables in the environment. PRM is more accurate and less diffusive than the piecewise linear method. Using the monotonic semi-Lagrangian scheme ensures nonnegative and nonoscillatory natural transport of materials such as cloud water, cloud ice, and aerosols. The calculation of transport using the semi-Lagrangian scheme is shown in  appendix E. This useful scheme is also employed in the calculation of vertical flux in the dynamical core of our global model (the vertically conservative semi-Lagrangian scheme; Yoshimura and Matsumura 2003, 2005; Yukimoto et al. 2011) and in the vertical grid transformation in the general-purpose coupler Scup (Yoshimura and Yukimoto 2008).

In the AS scheme in our model, the transport by compensatory subsidence is calculated with an explicit Eulerian scheme. Therefore, the mass-flux limiter is used to satisfy the CFL condition, but the limiter has a bad influence on model results (Lopez and Moreau 2005; ECMWF 2006) especially when a large time step is used and the intervals of the vertical levels are small. A semi-implicit scheme can be used to relax the CFL condition, but the implicit solution provides smoother and nonlocal vertical profiles of tendencies through its inherent diffusivity (ECMWF 2006). The semi-Lagrangian scheme described here relaxes the CFL condition and provides less diffusive profiles.

3. Model description

We use the Meteorological Research Institute atmospheric general circulation model (MRI-AGCM v3.2; Mizuta et al. 2012), which is based on the model jointly developed by the Japan Meteorological Agency (JMA) and the MRI (Mizuta et al. 2006). In MRI-AGCM v3.2, a two-time-level semi-implicit semi-Lagrangian scheme is used for time integration to allow a long time step. A vertically conservative semi-Lagrangian scheme (see section 2f) is adopted, and a correction method similar to that described by Priestley (1993) and by Gravel and Staniforth (1994) is used horizontally for global conservation of tracers. Some new physical parameterization schemes have been introduced as options into the model to make it more suitable for long-term simulations such as global warming experiments. The cumulus scheme options are the AS scheme used in the JMA operational model (Arakawa and Schubert 1974; Moorthi and Suarez 1992; Pan and Randall 1998; JMA 2007), the Kain–Fritsch scheme, and the new cumulus scheme described here. The cloud scheme options are the Smith scheme (Smith 1990) used in the JMA operational model, the Tiedtke scheme (Tiedtke 1993; Kawai 2006), and a two-moment bulk scheme newly developed especially for the MRI Earth System Model (Yukimoto et al. 2011). In this paper, we compare model simulations with the AS cumulus scheme and with the new cumulus scheme, using the Tiedtke cloud scheme in both cases.

4. Validity of the new cumulus scheme interpolating between two convective updrafts

To validate the new cumulus scheme, we compare results using the new cumulus scheme with those from an ensemble of runs of a Tiedtke-type scheme (a bulk mass-flux scheme with one convective updraft) using different entrainment rates. Each run of the Tiedtke-type scheme is performed by letting in the new cumulus scheme. We use the results of one time step runs on 1 July of the third year of the 7-yr run described in section 5. (In Figs. 35, we show the results at the grid point 9.828°N, 137.812°E, where there are convective updrafts with different heights due to different entrainment rates.)

Figure 3 shows the vertical distribution of the upward mass fluxes of the convective updrafts (see  appendix F for the calculation of the mass flux), the organized plus turbulent entrainment rates, and the turbulent detrainment rates. Figures 3a, 3d, and 3g show the results of 26 ensemble members, each of which has the different turbulent entrainment/detrainment rate where (i.e. , , …, or ). Here the results for , for example, are obtained by putting . The ensemble mean of the upward mass flux (Fig. 3a) is shown in Fig. 3b, which is calculated from

 
formula

where is the value of each member, is the number of members, and the weights of and (corresponding to and , respectively) are set to 0.5. Figures 3e and 3h show the largest and smallest values of the ensemble members at each levels for the entrainment rates (Fig. 3d) and the detrainment rates (Fig. 3g). Figures 3c, 3f, and 3i show the results of the new cumulus scheme with and . In Figs. 3d–i, turbulent entrainment and turbulent detrainment are enhanced near the cloud bottom [ and in Eqs. (12) and (13)]. In Figs. 3d–f, organized entrainment is large at lower levels and at about 400–500 hPa. The results in Figs. 3b and 3c, Figs. 3e and 3f, and Figs. 3h and 3i are very similar, indicating that the new cumulus scheme can represent the effect of the ensemble of multiple convective updrafts with different entrainment rates.

Fig. 3.

Vertical profiles of (a)–(c) the mass fluxes, (d)–(f) the organized plus turbulent entrainment rates, and (g)–(i) the turbulent detrainment rates. (a),(d),(g) The 26 ensemble members, each of which has the different entrainment/detrainment rate where (i.e., , , …, or ). (b) Ensemble mean of the mass fluxes of the 26 members. (e),(h) The largest and the smallest values of entrainment and detrainment rates of the ensemble members. (c),(f),(i) Results of the new scheme with and .

Fig. 3.

Vertical profiles of (a)–(c) the mass fluxes, (d)–(f) the organized plus turbulent entrainment rates, and (g)–(i) the turbulent detrainment rates. (a),(d),(g) The 26 ensemble members, each of which has the different entrainment/detrainment rate where (i.e., , , …, or ). (b) Ensemble mean of the mass fluxes of the 26 members. (e),(h) The largest and the smallest values of entrainment and detrainment rates of the ensemble members. (c),(f),(i) Results of the new scheme with and .

However, the upward mass flux of the new cumulus scheme in Fig. 3c is slightly larger in the upper troposphere and slightly smaller in the lower troposphere than that of the ensemble mean in Fig. 3b, and the largest entrainment rate and detrainment rate in Figs. 3f and 3i are slightly larger at the upper levels than those in Figs. 3e and 3h, which indicates that the cloud tops of convective updrafts in the new cumulus scheme are slightly higher than those in the ensemble mean.

This can be explained by Fig. 4, which shows the differences in the in-cloud virtual temperatures between the 26 members and the member with the smallest entrainment rate . The straight line at each vertical level is the linear interpolation between the value with the smallest entrainment rate and that with the largest entrainment rate. The values of the in-cloud virtual temperature obtained by the linear interpolation are close enough to those of the ensemble members, meaning that the linear interpolation is a good approximation. However the values of the linear interpolation are slightly larger than those of the ensemble members. That is, the virtual temperature obtained by the linear interpolation in the new cumulus scheme is slightly higher than that obtained by explicit calculation of the convective updraft with each entrainment rate. Therefore, the cloud tops of convective updrafts in the new cumulus scheme become slightly higher than those in the ensemble mean.

Fig. 4.

Differences in in-cloud virtual temperature (K) between 26 ensemble members with different entrainment rates and the member with the smallest entrainment rate at the vertical levels of 300, 500, 700, and 850 hPa. The straight line at each vertical level is the linear interpolation between the value with the smallest entrainment rate (the first member) and that with the largest entrainment rate (e.g., the 21st member at the 700-hPa level).

Fig. 4.

Differences in in-cloud virtual temperature (K) between 26 ensemble members with different entrainment rates and the member with the smallest entrainment rate at the vertical levels of 300, 500, 700, and 850 hPa. The straight line at each vertical level is the linear interpolation between the value with the smallest entrainment rate (the first member) and that with the largest entrainment rate (e.g., the 21st member at the 700-hPa level).

Figures 5a–d show the vertical distributions of the mass flux, the convective heating rate, the convective moistening rate, and the detrainment of cloud water, respectively. The results of the new cumulus scheme are compared with those of the ensemble means of 26 members, 6 members, and 2 members with equally spaced different entrainment rates. Here we suppose that an ensemble mean of an infinite number of members gives the right answer. There are 36 vertical levels below 100 hPa in the model used here, so the ensemble mean of 26 members is considered to be close enough to the right answer. The results of the new cumulus scheme are similar to those of the ensemble mean of 26 members in Figs. 5a–d. The differences are that the absolute values in the lower troposphere are slightly smaller and those in the upper troposphere are slightly larger in the new cumulus scheme compared with the ensemble mean of 26 members. This is because the cloud tops of convective updrafts in the new cumulus scheme are slightly higher than those in the ensemble mean of 26 members, as described above. The results of the ensemble mean of 2 members are very different from those of 26 members. The results of 6 members are similar to those of 26 members in Figs. 5b and 5c. However, the detrainment of the cloud water for six members (Fig. 5d) is too large at some vertical levels, indicating that an ensemble of six members is not sufficient to correctly evaluate the detrainment of cloud water.

Fig. 5.

Comparison of (a) the upward mass flux (), (b) the heating rate (), (c) the moistening rate (), and (d) the detrainment of cloud water content () between the ensemble means of convective updrafts and the new cumulus scheme. The red line is the ensemble mean of 26 convective updrafts, each of which has the entrainment/detrainment rate , , . . . , or . The green line is the ensemble mean of six convective updrafts, each of which has the entrainment rate , , . . . , or . The orange line is the ensemble mean of two convective updrafts, each of which has the entrainment rate or . The black line is the new scheme with the minimum entrainment rate and the maximum entrainment rate .

Fig. 5.

Comparison of (a) the upward mass flux (), (b) the heating rate (), (c) the moistening rate (), and (d) the detrainment of cloud water content () between the ensemble means of convective updrafts and the new cumulus scheme. The red line is the ensemble mean of 26 convective updrafts, each of which has the entrainment/detrainment rate , , . . . , or . The green line is the ensemble mean of six convective updrafts, each of which has the entrainment rate , , . . . , or . The orange line is the ensemble mean of two convective updrafts, each of which has the entrainment rate or . The black line is the new scheme with the minimum entrainment rate and the maximum entrainment rate .

5. Model climatology

Here we present the results of 7-yr simulations of the 60-km mesh (TL319L64) models using the AS scheme and the new cumulus scheme, and compare them with the observations. Climatological sea surface temperature (SST) and sea ice concentration of Reynolds and Smith (1994) are used as lower boundary conditions. The other boundary conditions are the same as those in the Atmospheric Model Intercomparison Project (AMIP)-type simulations presented in Mizuta et al. (2012), Murakami et al. (2012a,b), and Endo et al. (2012). In the AMIP-type simulations using the new cumulus scheme at the resolutions from TL95 (180 km) to TL959 (20 km), the resolution dependence of the results in the global-scale climate is small (Mizuta et al. 2012). In the AMIP-type simulations using the AS scheme, the maximum detrainment of cloud water content, which is used as a tunable parameter in the AS scheme, is increased to enhance precipitation in the western Pacific. Although the climatology of precipitation is improved by this tuning, a large warm bias in the tropical upper troposphere and a large wet bias in the tropical middle and upper troposphere appear. Therefore, this tuning is not used in the simulation using the AS scheme shown here.

Figures 6a and 6b show the annual mean precipitation climatology for the 7-yr simulations and the difference from the Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP; Xie and Arkin 1997) climatology. The statistics of bias, root-mean-square error (RMSE), and correlations (CORR) are better in the new scheme simulation than in the AS scheme simulation. Scores of the precipitation climatology in Asia including the Taylor diagram (Taylor 2001) are shown in Endo et al. (2012), and Taylor diagrams for precipitation, wind/height fields, and radiation climatology are shown in Mizuta et al. (2012), where the model using the new cumulus scheme obtains good results compared with other models. Figure 6c shows convective precipitation, and Fig. 6d shows large-scale condensation precipitation for the simulations. The difference between the AS scheme and the new cumulus scheme simulations is large in the tropics, where there is almost no large-scale condensation precipitation in the AS scheme simulation, whereas there is some in the new cumulus scheme simulation.

Fig. 6.

Annual mean climatology of 7-yr runs of the model using (left) the AS scheme and (right) the new scheme. (a) Precipitation (), (b) difference in precipitation between the models and CMAP (), (c) convective precipitation (), and (d) large-scale condensation precipitation (). In (b), statistics of bias, root-mean-square error (RMSE), and correlation (CORR) are also shown.

Fig. 6.

Annual mean climatology of 7-yr runs of the model using (left) the AS scheme and (right) the new scheme. (a) Precipitation (), (b) difference in precipitation between the models and CMAP (), (c) convective precipitation (), and (d) large-scale condensation precipitation (). In (b), statistics of bias, root-mean-square error (RMSE), and correlation (CORR) are also shown.

Figure 7 shows the annual-mean climatology of outgoing longwave radiation (OLR) and outgoing shortwave radiation (OSR) at the top of the atmosphere. The model results are compared with the Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) satellite dataset (Loeb et al. 2009). In the new scheme simulation, negative OLR bias and positive OSR bias appear around deep convection region in the tropics such as the intertropical convergence zone (ITCZ), while the bias is reversed in the AS scheme simulation. It has been verified that the positive OSR bias in the new scheme can be decreased by introducing the practical independent column approximation (PICA; Collins 2001; Nagasawa 2012), which improves the treatment of cloud overlap in shortwave radiation calculation. In the shallow convection regions in the subtropics, negative OLR bias and positive OSR bias are found in the AS scheme simulation. These are reduced in the new scheme, although negative OSR bias is enhanced along the west coast of the continents. It has also been verified that the negative OSR bias around these regions, as well as that in the Southern Ocean, can be reduced by including an estimation of inversion strength (EIS) by Wood and Bretherton (2006) into the stratocumulus scheme and modifying it to include the effect of cloud-top entrainment (CTE) instability (Kawai 2013).

Fig. 7.

As in Fig. 6, but for (a) outgoing longwave radiation (OLR), (b) difference in OLR between the models and the CERES-EBAF observation, (c) outgoing shortwave radiation (OSR), and (d) difference in OSR between the models and the CERES EBAF observation.

Fig. 7.

As in Fig. 6, but for (a) outgoing longwave radiation (OLR), (b) difference in OLR between the models and the CERES-EBAF observation, (c) outgoing shortwave radiation (OSR), and (d) difference in OSR between the models and the CERES EBAF observation.

Figure 8 shows the differences in zonal mean annual mean temperature and zonal mean specific humidity between the 7-yr simulations and the Japanese 25-year Reanalysis Project (JRA-25; Onogi et al. 2007). The warm bias in the tropical upper troposphere and the positive bias of water vapor in the tropical middle troposphere are smaller in the new cumulus scheme simulation than in the AS scheme simulation. The negative bias of water vapor at about 800–900 hPa in the subtropics is larger in the new cumulus scheme simulation. It has been verified that this bias can be decreased by weakening shallow convection. The shallow convection has the effect of drying at the low levels.

Fig. 8.

Zonal mean annual mean climatology of 7-yr runs of the model using (left) the AS scheme and (right) the new scheme. (a) Difference in temperature between the models and JRA-25 (). (b) Difference in specific humidity () between the models and JRA-25.

Fig. 8.

Zonal mean annual mean climatology of 7-yr runs of the model using (left) the AS scheme and (right) the new scheme. (a) Difference in temperature between the models and JRA-25 (). (b) Difference in specific humidity () between the models and JRA-25.

Figure 9a shows the simulated zonal mean annual mean convective updraft mass flux. The mass flux in the new cumulus scheme is larger in the lower troposphere and smaller in the upper troposphere than in the AS scheme, showing that there is more shallow/congestus convection and less penetrative convection in the new cumulus scheme than in the AS scheme. One of the reasons that there is more penetrative convection in the AS scheme is as follows: in the AS scheme in our model, the values of in-cloud variables at the cloud top are directly calculated from those at the cloud bottom without calculating conversion from cloud water/ice into precipitation at intermediate vertical levels, so freezing of cloud water is overestimated and the temperatures in the updrafts are overestimated. (However, this point is improved in the new AS-like scheme being developed in JMA.)

Fig. 9.

As in Fig. 8, but for (a) convective updraft mass flux (), (b) heating by cumulus scheme (), (c) moistening by cumulus scheme (), (d) heating by large-scale condensation scheme (), and (e) moistening by large-scale condensation scheme (). Negative regions are shaded in (d) and (e).

Fig. 9.

As in Fig. 8, but for (a) convective updraft mass flux (), (b) heating by cumulus scheme (), (c) moistening by cumulus scheme (), (d) heating by large-scale condensation scheme (), and (e) moistening by large-scale condensation scheme (). Negative regions are shaded in (d) and (e).

Figures 9b and 9c show the heating and moistening rates of the cumulus schemes. Corresponding to the difference in mass flux in Fig. 9a, heating and drying in the upper troposphere are larger in the AS scheme. The warm bias in the tropical upper troposphere in the AS scheme simulation seems to result from the large mass flux and heating in the upper troposphere. Figures 9d and 9e show the heating and moistening rates for the large-scale condensation scheme (Tiedtke cloud scheme). The simulations with the two cumulus schemes give substantially different results in the upper troposphere: positive heating and negative moistening rates in the new cumulus scheme simulation, and negative heating and positive moistening rates in the AS scheme simulation. This difference can be explained by the following equation in Tiedtke (1993):

 
formula

where is the saturation specific humidity, represents the compensatory subsidence (see  appendix F for the calculation of and ), represents the grid-scale vertical velocity, is the change in accompanying the change of pressure and temperature constrained to lie on a moist adiabat, and represents the change of temperature by radiative and turbulent processes, etc. In the typical convective regions in the tropics, there are grid-scale upward motions, so is negative. Since is positive, the sign of depends on whether or has the larger absolute value. Figure 10 shows the simulated June–July–August climatologies of and in the active regions of convections. In the new cumulus scheme simulation, negative is likely in the tropical upper troposphere because of small , which leads to from Eq. (44), and heating and drying by the large-scale condensation. In contrast, in the AS scheme simulation, is likely in the tropical upper troposphere due to large , which leads to , resulting in cooling and moistening by the evaporation of cloud water. It is difficult to say which is better; however, the results of the new cumulus scheme are closer to the following description in Johnson and Young (1983), “Mesoscale anvils, defined as widespread (~100 km) cloud systems extending from near the freezing level to the upper troposphere, are characterized by light stratiform precipitation,” and the descriptions in other papers (e.g., Leary and Houze 1979; Johnson 1984).

Fig. 10.

Longitude–vertical cross section averaged between 5° and 10°N of June–July–August climatology of the 7-yr runs using (left) the AS scheme and (right) the new scheme. (a) Compensatory subsidence multiplied by gravity acceleration, (). (b) Grid-scale vertical velocity plus (a), ().

Fig. 10.

Longitude–vertical cross section averaged between 5° and 10°N of June–July–August climatology of the 7-yr runs using (left) the AS scheme and (right) the new scheme. (a) Compensatory subsidence multiplied by gravity acceleration, (). (b) Grid-scale vertical velocity plus (a), ().

6. Model variability

The Wheeler–Kiladis diagrams (Wheeler and Kiladis 1999; Kim et al. 2009) are shown in Fig. 11, which shows zonal wavenumber–frequency power spectra of symmetric and antisymmetric components of OLR divided by the background power. The Advanced Very High Resolution Radiometer (AVHRR) OLR observation data (Liebmann and Smith 1996) are used to compare with the model results. The result of not only AVHRR 26-yr data from 1979 to 2005 but also 7-yr data from 1999 to 2005 is shown to compare 7-yr model results. The Madden–Julian oscillation (MJO) signal in the period of 30–80 days in the new cumulus scheme simulation is clearer than that in the AS scheme simulation, but their signals are still weaker than the observation. However, the latest version of MRI-AGCM (v3.3) used in MRI-CGCM3 (Yukimoto et al. 2012), where the new cumulus scheme with different tuning parameters and the two-moment bulk cloud scheme are used, shows good results in MJO predictions (N. P. Klingaman et al. 2014, manuscript submitted to J. Geophys. Res.; E. Shindo 2014, personal communication). The signal of the Kelvin waves in the AS scheme is larger than that of the new scheme, but their signals are weaker than the observation. The signals of equatorical Rossby (ER) waves in the AS and the new schemes are good compared with the observation. The signals of eastward inertia–gravity (EIG) waves in both schemes are very weak.

Fig. 11.

Wheeler–Kiladis diagrams: zonal wavenumber–frequency power spectra of symmetric and antisymmetric components of OLR divided by the background power, for (a) the 7-yr run using the AS scheme, (b) the 7-yr run using the new cumulus scheme, (c) the 27-yr AVHRR observation from January 1979 to December 2005, and (d) the 7-yr AVHRR observation from January 1999 to December 2005.

Fig. 11.

Wheeler–Kiladis diagrams: zonal wavenumber–frequency power spectra of symmetric and antisymmetric components of OLR divided by the background power, for (a) the 7-yr run using the AS scheme, (b) the 7-yr run using the new cumulus scheme, (c) the 27-yr AVHRR observation from January 1979 to December 2005, and (d) the 7-yr AVHRR observation from January 1999 to December 2005.

7. Estimation of computational costs of cumulus schemes

The whole computational time of the one-month runs of the models using AS scheme and the new scheme at the 60-km resolution (TL319L64) is about 11 480 and 11 280 s, respectively, when one node (16 processors of IBM Power 6) of Hitachi SR16000 supercomputer is used. The computational time of the AS scheme part and the new scheme part per one-month run is about 731 and 834 s, respectively, as shown in Table 1. When the AS scheme is used, the calculation of dynamic CAPE generation rate (DCAPE; Xie and Zhang 2000; Nakagawa 2008) and the horizontal advection of cloud-base mass fluxes are additionally executed, so the whole computational time of the one-month runs of the AS scheme model is larger than that of the new scheme model.

Table 1.

Computational time of the AS scheme and the new scheme per one-month run, and computational time of the M-mass-flux-type scheme roughly estimated from that of the new scheme, where is the number of the calculated updraft mass fluxes. Estimated time in cases of , , and is also shown. Values in brackets are the ratio of the time to that of the new scheme.

Computational time of the AS scheme and the new scheme per one-month run, and computational time of the M-mass-flux-type scheme roughly estimated from that of the new scheme, where  is the number of the calculated updraft mass fluxes. Estimated time in cases of , , and  is also shown. Values in brackets are the ratio of the time to that of the new scheme.
Computational time of the AS scheme and the new scheme per one-month run, and computational time of the M-mass-flux-type scheme roughly estimated from that of the new scheme, where  is the number of the calculated updraft mass fluxes. Estimated time in cases of , , and  is also shown. Values in brackets are the ratio of the time to that of the new scheme.

The computational time of other types of cumulus schemes are also estimated in Table 1. In the computational time of the new scheme, the time of the part where the computational time becomes twice by calculating two convective updraft mass fluxes and that of the rest part are about 50/50. Therefore, when the number of the multiple updraft mass fluxes are explicitly calculated, the computational time are roughly estimated to become about times as long as that of the new scheme as shown in Table 1. The computational time in the Tiedtke-type schemes is estimated to be about 0.75 times as long as that in the new scheme by substituting . The computational time in the cumulus scheme explicitly calculating almost all (some) updraft mass fluxes layer by layer is roughly estimated to be 7 (3) times as long as that in the new scheme, for example, when ().

8. Summary and conclusions

We have developed a new cumulus scheme that explicitly represents multiple cumulus clouds with different cloud tops by interpolating between two convective updrafts with large and small entrainment rates. The two convective updrafts are explicitly calculated and the variables (e.g., temperature and specific humidity) in convective updrafts with intermediate entrainment rates are obtained by linear interpolation. The new scheme has the advantage of the AS-type scheme in that the effects of multiple cumulus clouds are represented explicitly, and the advantage of the Tiedtke-type scheme in that in-cloud variables are calculated layer by layer.

In this scheme, a conservative and monotonic semi-Lagrangian scheme with the PRM interpolation profile is adopted to calculate the vertical transport by convection-induced compensatory subsidence. This relaxes the mass-flux limit due to the CFL condition, and ensures nonnegative natural material transport.

The results of this scheme are equivalent to those of a 26-member ensemble mean, in which each member has one of 26 equally spaced different entrainment rates. The new scheme, whose computational cost is much lower than that of the explicit 26-member calculation, can produce equivalent results. Rough estimation of the computational time of the cumulus scheme explicitly calculating 26 convective updrafts layer by layer is about 7 times as that of the new scheme.

The climatologies of precipitation distribution and zonal mean temperature in the model using the new scheme are closer to the observations than in the model using the AS scheme. In the AS scheme, the mass flux in the tropical upper troposphere is larger than in the new scheme, which seems to be related to the warm bias there.

The simulation of MJO in the new scheme is slightly better than in the AS scheme, and the latest version of MRI-AGCM shows good results in MJO predictions. The signals of EIG waves in both schemes are very weak. We will try to improve the variability by modifying entrainment or closure assumption, etc., by referring to the papers describing entrainment (e.g., de Rooy et al. 2013) and the cumulus schemes showing good results in variability (e.g., Bechtold et al. 2008; Chikira and Sugiyama 2010).

Acknowledgments

We are grateful to Masato Sugi, Tomoaki Ose, Seiji Yukimoto, Yoshinori Yamada, Eiki Shindo, MRI model development members, and MRI model analysis members for providing important information about cumulus parameterization and MJO, evaluating the cumulus scheme, and encouraging us. This work was conducted under the framework of the “Development of Basic Technology for Risk Information on Climate Change” supported by the SOUSEI Program of the Ministry of Education, Culture, Sports, Science, and Technology of Japan. Figure 11 was made with a diagnostic tool developed by the U.S. CLIVAR MJO Working Group.

APPENDIX A

Discretization of Convective Updrafts

Variables such as , , , , and are located at the full levels ; is the lowest full level. The half level is located at the boundary between the full levels and .

Equation (17) is discretized as

 
formula

where . Equation (A1) is calculated by dividing it into two steps as follows:

 
formula
 
formula

The relation between and and the mass flux of each cloud type is shown in Fig. 2b. The variable is calculated from

 
formula

where is obtained from Eq. (A11) below.

Equation (2) is discretized as

 
formula
 
formula
 
formula

In Eq. (A7), is obtained by linear interpolation of and (see Fig. 2a). The variables , , , , , , , , and are calculated from the equations obtained by discretizing Eqs. (3)(5) similarly. Condensation ( and ) occurs when convective updrafts are saturated.

The coefficients and in Eqs. (12) and (13) are given by

 
formula

where is the lowest vertical level of condensation and is the lowest vertical level of positive buoyancy. The values of and are given in Eqs. (14) and (15). We use the values

 
formula
 
formula

Equation (A12) below gives . The vertical distributions of and are shown in Fig. 2a.

The virtual temperatures in convective updrafts with entrainment rates between and are assumed to be obtained by linear interpolation of and . When the virtual temperatures in a part of the convective updrafts fall below the environmental virtual temperature , that part of the convective updrafts is detrained as organized detrainment (see Fig. 2). The proportion of detrainment is obtained by

 
formula

The position of between and , , is obtained from

 
formula

and , , , and in the organized detrainment are obtained from

 
formula

where and .

APPENDIX B

Discretizing the Closure Assumption

To obtain in Eq. (26), and , which are the CAPE of cumulus [a] and cumulus [b] up to the vertical level k, respectively, are calculated from

 
formula

is obtained from

 
formula

The decrease in due to the convective updraft of the cloud type is obtained from

 
formula

where is the vertical integral up to the vertical level .

The variable is defined as the provisional upward mass flux when it is assumed that organized detrainment does not occur. Then, is obtained from

 
formula

(see Fig. 2b). Organized entrainment is independent of cloud type , and the turbulent entrainment and the turbulent detrainment have the same value and do not affect the magnitude of the mass flux. So, in Eq. (21) satisfies

 
formula

not only at the level , but also at the vertical levels . It is assumed to be satisfied at the levels (see Figs. 2a,b). From Eqs. (22), (26), (B3), and (B5),

 
formula

is obtained.

Final (not provisional) values are obtained as follows. The organized detrainment is obtained from

 
formula

while and are calculated from the top level to the lower levels using the recurrence relation:

 
formula
 
formula
 
formula

The variables , , , and , which are proportional to , are also obtained by multiplying with the provisional values as in Eq. (B9).

APPENDIX C

Discretization of the Convective Downdraft

Equations (27) and (28) are discretized, respectively, as

 
formula
 
formula

Equation (29) is discretized in the same way as Eq. (28).

APPENDIX D

Discretization of Feedback to Environment

When calculating the effect of cumulus convection on the environment, the ensemble mean of the convective updrafts of the different cloud types, rather than the value of each cloud type , is necessary. Hereafter, is the ensemble mean of , and is calculated from

 
formula

where is calculated from the recurrence relation:

 
formula

Equation (D2) for the case is derived by comparing the equation

 
formula

with the equation obtained by substituting Eqs. (D1), (A13), and (A7) (where is replaced with ) into

 
formula

The variables and are obtained from

 
formula
 
formula

The ensemble means of the detrained properties from the convective updrafts of all cloud types, , , , and , are obtained from

 
formula

where or .

Only Eq. (39) is discretized below, but Eqs. (40)(42) are discretized in the same way. Considering that is the average of and , and using Eqs. (D5)(D7), Eq. (39) is discretized as

 
formula

where is the value of after time integration with the cumulus scheme, and is the time step. By using the following definitions:

 
formula
 
formula
 
formula
 
formula

Eq. (D8) is transformed into

 
formula

The second and the third terms in Eq. (D13) represent the vertical flux by compensatory subsidence.

APPENDIX E

Semi-Lagrangian Calculation of Transport by Compensatory Subsidence

Figure D1 shows the vertical transport of mass along with the compensatory subsidence. In Fig. D1,

 
formula

is the mass (multiplied by ) of the environment excluding the entrainment by cumulus convection, while

 
formula

is the mass of the environment excluding the detrainment from cumulus convection. The vertical summation of the entrainment is equal to the vertical summation of the detrainment, so

 
formula

is satisfied. The gray area in the central part of Fig. D1 represents , the mass moving from the upper layer to the lower layer associated with the compensatory subsidence. Figure D2 shows the vertical flux of the variable (such as and ). Here, in the left panel of Fig. D2 is the value before the vertical flux calculation. Using the piecewise rational method (PRM), an interpolation function is obtained, where the mean value of in layer before the vertical flux calculation agrees with . Next, is obtained by calculating the mean value of in layer after the vertical flux calculation (see the right panel of Fig. D2). The gray area in the central part of Fig. D2 represents , the quantity of moving from the upper layer to the lower layer , and

 
formula

is satisfied. Here the conservation equations over the column

 
formula

are satisfied. From , and using Eqs. (E4), (E1), and (E2), Eq. (D13) is transformed to

 
formula

from which is obtained.

Fig. D1.

Vertical transport of mass by compensatory subsidence. The gray area in the middle figure shows the mass moving from the upper layer to the lower layer.

Fig. D1.

Vertical transport of mass by compensatory subsidence. The gray area in the middle figure shows the mass moving from the upper layer to the lower layer.

Fig. D2.

Vertical transport of static energy, water vapor, etc., by compensatory subsidence. The gray area of the middle figure shows the quantity moving from the upper layer to the lower layer.

Fig. D2.

Vertical transport of static energy, water vapor, etc., by compensatory subsidence. The gray area of the middle figure shows the quantity moving from the upper layer to the lower layer.

APPENDIX F

Values of Mass Flux at Half Levels and at Full Levels

The upward mass flux at the half level is given by

 
formula

because is the mass moving from the full level to the full level per unit time [see Eq. (A2)].

The downward mass flux in the convective downdraft at the half level is given by

 
formula

because in Eq. (C1) is the mass moving from the full level to the full level per unit time.

The mass fluxes at the full level in Eq. (44) and in Figs. 3, 5, and 9 are obtained from

 
formula
 
formula

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