1. Introduction
The importance of modeling the multiscale effects of moist convection was recognized around 1960, when the early attempts to numerically simulate tropical cyclone development failed to produce a large-scale vortex [see Kasahara (2000) for a review]. Subsequently, the idea of cumulus parameterization emerged and its primitive forms were introduced in tropical cyclone modeling (Charney and Eliassen 1964; Ooyama 1964) and, quite independently, in general circulation modeling (Manabe et al. 1965). Since then, cumulus parameterization has always been a central issue in numerical modeling of the atmosphere [see Arakawa (2004) for a review]. In spite of this, the rate of progress during the last several decades has been unacceptably slow, especially when it is compared to the rapid expansion of the scope of general circulation models (GCMs); so we may say, “the cloud parameterization problem is deadlocked” (Randall et al. 2003).
One of the most important sides of the deadlock is that the existing atmospheric models represent the complexities of cloud systems only in one of the following two ways: highly parameterized and explicitly simulated. The former has the objective of representing “apparent sources” (Yanai et al. 1973) that include the effect of subgrid vertical eddy transports, while the latter has that of representing “true sources” that are taking place locally. Correspondingly, besides those models that explicitly simulate turbulence, there are two discrete families of atmospheric models, as shown in Fig. 1a: one is represented by the conventional GCMs and the other by the cloud-resolving models (CRMs). In this figure, the abscissa is the horizontal resolution and the ordinate is a measure of the degree of parameterization, such as reduction in the degrees of freedom, increasing downward. These two families have been developed for applications to quite different ranges of the horizontal resolution in mind. What we see here is a polarization of atmospheric models separated by the “gray zone” in the mesoscale range. The difficulties in representing moist convective processes in this range have long been recognized among the mesoscale modeling community (e.g., Molinari and Dudek 1992; Molinari 1993; Frank 1993).
The dipolarization and tripolarization of atmospheric models. See the text for further explanation.
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
Recently, a new family of global models called the Multiscale Modeling Framework (MMF) has been added to the two families, as shown in Fig. 1b. The MMF is a coupled GCM–CRM system that follows the “super parameterization” approach introduced and used by Grabowski and Smolarkiewicz (1999), Grabowski (2001), and Khairoutdinov and Randall (2001). In this approach, cumulus parameterization in conventional GCMs is replaced with the mean effects of cloud-scale processes simulated by a 2D CRM embedded in each GCM grid cell. In this way, the CRM physics is used while staying with the standard coarse resolution for the GCM.
Because of the recent advancement of computer technology, straightforward applications of 3D nonhydrostatic models with explicit cloud microphysics have begun to be feasible even for studying the global climate (e.g., Sato et al. 2009). When a sufficiently high horizontal resolution is used, these models become global CRMs (GCRMs), as shown in Fig. 1b. This is an exciting development in the history of numerical modeling of the global atmosphere. The result is, however, a tripolarization of global models with the model physics formulated quite differently from each other. Because of this, both the GCMs and the MMF do not converge to a GCRM as the horizontal grid spacing is reduced even when they share the same dynamics core. The lack of convergence is not scientifically healthy since the spectrum in between the two scales represented by GCMs and GCRMs can be virtually continuous because of the existence of mesoscale phenomena. For such a spectrum, the use of discretized equations can be justified only when the error resulting from the discretization can be made arbitrarily small by using higher resolutions.
Jung and Arakawa (2004) showed convincing evidence for the transition of model physics as the resolution changes by performing budget analysis of data simulated by a CRM. In this analysis, a high-resolution run with cloud microphysics is made first (control). Lower-resolution test runs are then made without cloud microphysics over a selected time interval to identify the contributions from the dynamics core. By comparing these results with those of the control, the apparent cloud microphysical source “required” for the low-resolution solution to become equal to the space/time averages of the high-resolution solution is identified. This procedure was repeated over many realizations selected from the control. Figure 2a shows examples of the domain- and ensemble-averaged profiles of the required source of moist static energy obtained in this way. Here moist static energy is defined by 
Domain- and ensemble-averaged profiles of the “required source” for (a) moist static energy (divided by 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
Low-resolution models such as the conventional GCMs are supposed to produce profiles similar to the green profiles shown in Fig. 2, which may be called the GCM type, while high-resolution models such as the CRMs are supposed to produce profiles similar to the red profiles, which may be called the CRM type. As Arakawa (2004) emphasized, any space/time/ensemble average of CRM-type profiles does not give a GCM-type profile. This means that the cumulus parameterization problem is more than a statistical theory of cloud microphysics. Also, it is not a purely physical/dynamical problem because it is needed as a consequence of mathematical truncation. Finally, it is not a purely mathematical problem since the use of a higher resolution while using the same formulation of model physics does not automatically improve the result (e.g., Williamson 1999; Buizza 2010). A complete theory for cumulus parameterization must address all of these aspects in a consistent manner, including the transition between the GCM-type and CRM-type profiles.
As we have seen in Fig. 2, the required sources highly depend on the horizontal resolution of the model, as well as on the time interval for implementing model physics. A model that unifies low- and high-resolution models must produce this kind of transition when applied to different resolutions. Obviously, such a model must have a unified dynamics core, which is necessarily nonhydrostatic, and a unified formulation of model physics applicable to all resolutions. The former is rather straightforward, but unfortunately, the latter is not. Conventional cumulus parameterization schemes cannot be used for the unified formulation because they assume either explicitly or implicitly that the horizontal grid spacing and the time interval for implementing physics are sufficiently larger and longer than the size and lifetime of individual moist convective elements.
Arakawa et al. (2011) (and also Arakawa and Jung 2011) discussed two routes to unify these families of atmospheric models, shown as route I and route II in Fig. 3. The departure points of route I and route II are the conventional GCMs and a new generation of MMF, respectively, but they share the same destination point, GCRM. Route I breaks through the gray zone by generalizing a conventional cumulus parameterization in such a way that it converges to an explicit simulation of deep moist convection. On the other hand, route II bypasses the gray zone by replacing the conventional cumulus parameterization with explicit simulations of deep moist convective processes regardless of the horizontal resolution of the GCM. On this route, the apparent sources are not explicitly formulated but are expected to be numerically simulated.
Two routes for unifying the low- and high-resolution models.
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
The purpose of this paper is to discuss a generalization of the conventional cumulus parameterizations following route I, while an approach following route II called the quasi-3D (Q3D) MMF is presented by Jung and Arakawa (2010) and will be updated in their forthcoming paper. The route I approach, which effectively unifies representations of deep moist convection in GCMs and CRMs, is called “unified parameterization.” Since the introduction of this subject by Arakawa et al. (2011), the algorithm has been made more elaborate and complete as we discuss in this paper. Section 2 identifies the problems to be addressed in developing a unified parameterization and introduces the fractional convective cloudiness σ as a key parameter. Section 3 discusses the dependence of vertical transports on σ and its parameterization. Section 4 presents a closure of the parameterization including the determination of σ for each realization of the grid-scale processes, while section 5 discusses the problem of parameterizing uncertainties in the framework of unified parameterization. Finally, summary and further discussions are presented in section 6. Remaining problems including the vertical structure of eddy transports and the dependence of cloud microphysics and dynamics on σ will be discussed in Part II of this paper (A. Arakawa and C.-M. Wu 2013, unpublished manuscript).
2. Identification of the problems
The first step to open route I is reexamination of the widely used assumption that convective updrafts cover only a small fraction of a GCM grid cell. Most conventional cumulus parameterizations that explicitly use a cloud model (e.g., Arakawa and Schubert 1974; Emanuel 1991; Gregory and Rowntree 1990; Kain and Fritsch 1990; Tiedtke 1989; Zhang and McFarlane 1995) assume this, at least implicitly, regarding the temperature and water vapor mixing ratio carried by GCM grid points as if they represent those of the cloud environment. Then, as shown by Arakawa and Schubert (1974), the processes to be considered for predicting those variables are the “cumulus-induced” subsidence in the environment and the detrainment of cloud air into the environment. (Here it is important to recognize that the cumulus-induced subsidence is a component of the hypothetical subgrid-scale eddy circulation, which is closed within the same grid cell by definition. The true subsidence is a component of the total circulation, which is the sum of the subgrid- and grid-scale circulations.)
Let σ be the fractional convective cloudiness, which is the fractional area covered by convective updrafts in the grid cell. When the area covered by individual updrafts is fixed, σ represents the population of updrafts in the grid cell. (The fractional convective cloudiness defined in this way should be distinguished from the more commonly used fractional cloudiness that matters for radiation.) In terms of σ, the assumption mentioned above means 
To visualize the problem to be addressed, we have performed two numerical simulations using a CRM, one with and the other without background shear. The model used for these simulations is the 3D vorticity equation model of Jung and Arakawa (2008) with a three-phase cloud microphysics parameterization (Krueger et al. 1995) applied to an idealized horizontally periodic domain over ocean with a fixed surface temperature. The horizontal domain and grid spacing are 512 and 2 km, respectively. Other experimental settings follows the benchmark simulations performed by Jung and Arakawa (2010), in which the thermodynamic forcing is prescribed in terms of the vertical profiles of mean cooling and moistening rates, chosen to counteract the apparent heat source and moisture sink typical of the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) phase III after some idealization. The constant cooling rate of 2 K day−1 is also included to mimic radiative cooling. In the simulation with the background shear, wind components are nudged to prescribed profiles representing a typical Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) condition with a 2-h time scale.
Figure 4 shows snapshots of the vertical velocity w at 3-km height simulated with and without background shear. As we can see from these snapshots, these two runs represent quite different cloud regimes. To see the grid-scale dependence of the statistics, we divide the original CRM domain (512 km × 512 km) into subdomains of identical size d to represent the GCM grid cells. An example of the subdomains with d = 64 km is shown in the figure.
Snapshots of the vertical velocity w at 3-km height simulated (a) with and (b) without background shear, and an example of the subdomains to show the grid-spacing dependence of the statistics.
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
In the following diagnosis of these data, σ is determined for each subdomain by the fractional number of CRM grid points that satisfy 
The resolution dependence of 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
We now look into the resolution dependence of the vertical transport of moist static energy in updraft-containing subdomains. (We have found that the vertical transport in subdomains with no updrafts is negligible.) This transport may be written as 
The diagnosed total and eddy transports of moist static energy divided by 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
3. The σ dependence of vertical transport and its parameterization
a. Diagnosed σ dependence of the vertical transport
Section 5 will show that the standard deviation associated with the ensemble-average transport shown in Fig. 6 is quite large, indicating that there are a variety of situations even when the resolution is fixed. To obtain an insight into the factor controlling the magnitude of vertical transports, we classify subdomains of the same size into different bins according to the values of σ. Figure 7 shows the σ dependence of the ensemble-mean vertical transport of moist static energy obtained in this way for the shear case. The case of 
The σ dependence of the ensemble-mean total (red) and eddy (green) vertical transports of moist static energy divided by 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
Figure 7 shows that the eddy transport does not vanish even for 
Figure 8 is the same as Fig. 7, but for 
As in Fig. 7, but for (a) 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
To see the situation for different values of d, Fig. 9 presents the ratio 
The ratio 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
b. Parameterization of the σ dependence of eddy transport by homogeneous updrafts/environment
We now consider the problem of parameterizing the σ dependence of the vertical eddy transport. In addition to the assumption 
1) Conventional parameterizations
2) Unified parameterization
c. Diagnosed eddy transport by homogeneous updrafts and environment
To see the σ dependence of the statistics when the top-hat profile assumption is used, we modify the data by replacing w and thermodynamic variables of all updraft points in each subdomain by their averages, and do the same for the environment points. We first check whether 
Diagnosed 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
The light blue lines in Figs. 7 and 8 show the eddy transports diagnosed from the modified dataset. As anticipated, these lines are very close to the curve 
In Fig. 11a, which partially reproduces Fig. 7, the blue dashed curve shows 
The red and light blue lines are as in Fig. 7 for horizontal resolutions of (a) 8, (b) 16, and (c) 32 km. The blue dashed lines show the best-fit 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
So far we have seen the σ dependence of diagnosed vertical transports at the 
4. Closure of the parameterization and determination of σ
In this section, we first review the closure assumption typically used in the conventional parameterizations and then discuss the closure of the unified parameterization including determination of the fractional convective cloudiness σ.
a. Conventional parameterization with full adjustment
For a review of the closure assumptions used in the conventional parameterizations, see Arakawa (2004) and references therein. Most parameterizations currently being used are adjustment schemes, in which a vertically integrated measure of convective instability such as CAPE or cloud work function (Arakawa and Schubert 1974) is adjusted toward its equilibrium value. Typically the adjustment is relaxed using a somewhat arbitrarily chosen finite time scale (e.g., Moorthi and Suarez 1992).
b. Unified parameterization: Reduction of the vertical eddy transports
c. Determination of σ
Plot of σ given by (19) as a function of λ.
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
The unified parameterization uses the value of σ determined in this way in (16), not only for h but also for other ψ, assuming that the variables other than h play only passive roles as far as the process of controlling σ is concerned. Verifying this assumption, however, requires choosing a particular cloud model, which is beyond the scope of this paper.
d. Physical meaning of λ
We find in Fig. 13 that 
Plots of the ensemble averages of 
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
Recall that 
e. Outline of the procedure in practical applications
Since the unified parameterization is an approach following route 1 shown in Fig. 3, we require that it reduce to an existing parameterization scheme with full adjustment as 
The plume model in the parameterization scheme can also be used to determine 
5. Remark on parameterization of uncertainty
Most existing cumulus parameterization schemes attempt to deterministically formulate the large-scale effects of cumulus convection. There is no reason to believe, however, those effects can be determined uniquely from the grid-scale variables predicted by the model. It is then obvious that we should introduce stochastic effects in one way or another at some point in the development of cumulus parameterization. It should be remembered, however, that parameterization is not a purely statistical problem as pointed out in section 1, and thus stochastic formulation must be made under appropriate physical, dynamical, and computational constraints that determine the major source of uncertainty. When viewed from this point, there are considerable differences among the stochastic parameterizations so far proposed (e.g., Ball and Plant 2008; Buizza et al. 1999; Lin and Neelin 2000, 2002, 2003; Majda and Khouider 2002; Khouider and Majda 2007; Neelin et al. 2008; Palmer et al. 2005; Plant and Craig 2008; Shutts and Palmer 2007).
As in Fig. 6, Fig. 14 shows the resolution dependence of the diagnosed vertical transports 
As in the shear case of Fig. 6, but with standard deviations for the total and eddy transports.
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
As long as 
6. Summary and further discussion
One of the most important issues to be addressed in multiscale modeling of the moist-convective atmosphere is that the existing atmospheric models represent the effects of deep moist convection only in one of the following two ways: highly parameterized as in conventional general circulation models (GCMs) and explicitly simulated as in cloud-resolving models (CRMs). Correspondingly, numerical models of the atmosphere are polarized into two families represented by the GCMs and CRMs, separated by the gray zone for the mesoscale range. This paper presents a new framework for cumulus parameterization applicable to any horizontal resolution between those typically used in GCMs and CRMs. For sufficiently low resolutions, the framework is equivalent to the use of a conventional parameterization with full adjustment to a quasi-equilibrium state. For sufficiently high resolutions, on the other hand, it reduces to an explicit simulation of deep moist convective processes as is done in CRMs. Since parameterizations in GCMs and CRMs are unified in this way, we call the framework unified parameterization.
It is emphasized that a cumulus parameterization is supposed to formulate only the subgrid effects of cumulus convection, not the total effects involving grid-scale motion. Then the transport to be parameterized is only the eddy transport, not the total transport. The unified parameterization formulates the eddy transport in such a way that a smooth transition between the two ways of representing deep moist convection can take place. The key parameter to allow this transition is σ, which is the fractional area covered by convective updrafts in the grid cell, rather than the resolution itself. Practically all conventional cumulus parameterizations assume 
The unified parameterization formulates this transition by eliminating the assumption of 
The unified parameterization can also provide a framework for including stochastic parameterization. It is pointed out that a stochastic formulation must be made under appropriate physical, dynamical, and computational constraints that identify the source of uncertainty. In the unified parameterization, the source is in the determination of cloud properties relative to the gridpoint values, which influences the uncertainty of σ and hence that of eddy transports. We suspect that different phases of cloud development are primarily responsible for the uncertainty, which could be formulated stochastically.
The remaining issues include parameterization of the eddy transport because of the inhomogeneous structure of updrafts and the environment, which is responsible for the difference between the green and light blue lines shown in Figs. 7 and 8. The eddy transports shown in light blue are obtained from the modified dataset, in which w and thermodynamic variables of all CRM points in the subdomain that satisfy 
As in Fig. 7, but with additional results based on double and triple internal structures of updrafts.
Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0330.1
The remaining issues also include the effects of convective downdrafts. For the datasets we have been using, it is found that the effect of downdrafts on the vertical eddy transport of moist static energy is rather small. For example, Fig. 15 shows that considering the triple internal structures of updrafts while staying with the homogeneous environment is almost sufficient to recover the green curve. This is probably because the characteristic magnitude of w for downdrafts is relatively small compared to that of active updrafts, and also the difference of moist static energy between downdrafts and the environment is typically small. This does not mean, however, that other effects of downdrafts, such as cooling and moistening resulting from the evaporation of precipitation in the environment, are also small. The unified parameterization allows inclusion of these effects determined by a conventional parameterization, but with their own σ dependence shown in Part II of this paper (A. Arakawa and C.-M. Wu 2013, unpublished manuscript).
Finally, it should be pointed out that the CRM-simulated datasets we have used for diagnosis by no means fully represent nature. The horizontal resolution of 2 km is almost certainly too coarse. Moreover, the CRM uses a highly simplified turbulence parameterization (first order) and an idealized microphysics parameterization. Thus, any diagnostic results presented here should not be taken as the final words. We believe, however, the strategy discussed in this paper is a reasonable step toward a truly unified cloud parameterization.
In this paper, we have shown only vertical transports diagnosed at a particular height. The vertical structure of the vertical transports as well as that of the horizontal transports will be presented in Part II (A. Arakawa and C.-M. Wu 2013, unpublished manuscript). Part II also discusses other topics including the dependence of the cloud-scale physical and dynamical effects on σ.
Acknowledgments
We wish to thank Professor David Randall for his interest and support of this work and for many valuable comments on the manuscript. We also thank Professor Steven Krueger and Dr. Joon-Hee Jung for their careful reviews of the original manuscript and for a number of suggestions for improvement. Constructive comments by three anonymous reviewers are also appreciated. The first author is supported by the National Science Foundation Science and Technology Center for Multi-Scale Modeling of Atmospheric Processes, managed by Colorado State University under Cooperative Agreement ATM-0425247. The second author is supported by Taiwan's National Research Council through Grant 101-2111-M-002-006 to National Taiwan University.
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