## 1. Introduction

Organized convection in the tropics involves a hierarchy of temporal and spatial scales ranging from short-lived mesoscale organized squall lines on the order of hundreds of kilometers to intraseasonal oscillations over planetary scales on the order of 40 000 km (Nakazawa 1974; Hendon and Liebmann 1994; Wheeler and Kiladis 1999). Despite the continued research efforts by the climate community, the present coarse-resolution GCMs, used for the prediction of weather and climate, poorly represent variability associated with tropical convection (Slingo et al. 1996; Moncrieff and Klinker 1997; Scinocca and McFarlane 2004; Lau and Waliser 2005; Zhang 2005). It is believed that the deficiency is due to inadequate treatment of cumulus convection (Moncrieff and Klinker 1997; Lin et al. 2006). Given the importance of the tropics for short-term climate and medium- to long-range weather prediction, the search for new strategies for parameterizing the unresolved effects of tropical convection has been the focus of researchers during the last few decades.

The search for methods of adequately addressing the interactions across temporal and spatial scales between the large-scale circulation and organized cloud systems, from individual clouds to large-scale clusters and superclusters to planetary-scale disturbance, has not been fruitless. Cloud-resolving models on fine computational grids and high-resolution numerical weather prediction models with improved convective parameterizations succeed in representing some aspects of organized convection (Grabowski 2003; Moncrieff et al. 2007). However, because of their extremely high computational cost, these methods cannot be applied to large ensemble-size weather prediction or climate simulations. The complexity of the problem has motivated the development of an approach that directly addresses the multiscale nature of the problem. Superparameterization (SP) methods (Grabowski and Smolarkiewicz 1999; Grabowski 2001, 2004; Randall et al. 2003; Majda 2007) use a cloud-resolving model (CRM) in each column of the large-scale GCM to explicitly represent small-scale and mesoscale processes and interactions among them. The computational cost can be further reduced by techniques such as sparse space–time SP (Xing et al. 2009). Nonetheless, computationally inexpensive GCM parameterizations that capture the variability and coherent structure of deep convection have remained a central unsolved problem in the atmospheric community.

The most common conventional cumulus parameterizations are based on the quasi-equilibrium (QE) assumption first postulated by Arakawa and Schubert (1974), the moist convective adjustment idea of Manabe et al. (1965), or large-scale moisture convergence closure of the Kuo (1974) type. As such, the mean response of unresolved modes on large-scale–resolved variables is formulated according to a prescribed deterministic closure. While many recipes for the closures have been created (Kain and Fritsch 1990; Betts and Miller 1986; Zhang and McFarlane 1995), these purely deterministic parameterizations were found to be inadequate for the representation of the highly intermittent and organized tropical convection (Palmer 2001). Many of the improvements in GCMs of the last decade came from the relaxation of the QE assumption, for example, through the addition of a stochastic perturbation. Buizza et al. (1999) used a stochastic backscattering model to represent the model uncertainties in a GCM; Lin and Neelin (2003) used a stochastic parameterization to randomize the way in which deep convection responds to large fluctuations via a prescribed probability distribution function for the convective time scale. Majda and Khouider (2002) were the first to propose a stochastic model for convective inhibition (CIN) that allows both internal interactions between convective elements and two-way interactions between the convective elements and the large-scale–resolved variables. Their model is based on an Ising-type spin–flip model used as a model for phase transitions in material science (Katsoulakis et al. 2003a). When coupled to a toy GCM, this stochastic parameterization produced eastward propagating convectively coupled waves that qualitatively resemble observations (Khouider et al. 2003; Majda et al. 2008) despite the extreme simplicity of the model and deficiency of the underlying convective parameterization. Stochastic processes have been used to parameterize convective momentum transport (Majda and Stechmann 2008) and to improve conceptual understanding of the transition to deep convection through critical values of column water vapor (Stechmann and Neelin 2011), as well as for the analysis of cloud cover data in the tropics and the extratropics (Horenko 2011).

The stochastic multicloud model for tropical convection introduced by Khouider et al. (2010, hereafter KBM10) is a novel approach to the problem of missing tropical variability in GCMs. The stochastic parameterization is based on a Markov chain lattice model where each lattice site is either occupied by a cloud of a certain type (congestus, deep, or stratiform) or it is a clear sky site. The convective elements interact with the large-scale environment and with each other through convective available potential energy (CAPE) and middle troposphere dryness. When local interactions between the lattice sites are ignored, a coarse-grained stochastic process that is intermediate between the microscopic dynamics and the mean field equations (Katsoulakis et al. 2003a; Khouider et al. 2003; Majda et al. 2008) is derived for the dynamical evolution of the cloud area fractions. Besides deep convection, the stochastic multicloud model includes both low-level moisture preconditioning through congestus clouds and the direct effect of stratiform clouds including downdrafts that cool and dry the boundary layer. The design principles of the multicloud parameterization framework are extensively explored in the deterministic version of the model developed by Khouider and Majda (2006a,b, 2007, 2008a,b; hereafter KM06a, KM06b, KM07, KM08a, and KM08b, respectively).

Here a slightly modified version of the stochastic multicloud model, coupled to simplified primitive equations, with vertical resolution reduced to the first two baroclinic modes, is used to study flows above the equator without rotation effects. The major modifications to the basic stochastic multicloud model (KBM10) include direct dependence of congestus cloud cover on low-level CAPE, a simplified stratiform heating closure, and inclusion of stratiform rain in the precipitation budget. Additionally, minor changes in parameters, motivated in part by the single column sensitivity studies in KBM10, are introduced to further improve the intermittency of the solutions. The single column analysis of KBM10 is extended to flows above the equatorial ring. In this paradigm setting, we illustrate the advantages of the stochastic model through side by side comparison with deterministic parameterizations with clear deficiencies. Additional emphasis is placed on natural adaptability of the model to moderate and coarse GCM resolutions.

A brief review of the most salient features of the stochastic multicloud model along with major modifications in closures is presented in section 2. The section also introduces a benchmark deterministic parameterization with clear deficiencies used here to illustrate the advantages of stochastic parameterization. In section 3, single column simulation results are used to validate the changes in the tuning parameters of the stochastic parameterization. Additionally, the section contains a study of sensitivity of the new parameterization (in single column mode) to changes in the SST, which here is deemed the “mock” warm pool. In section 4, the parameterization is used to study flows above the equator on a moderate-size GCM grid (40 km) in both aquaplanet and SST gradient regimes. Meanwhile, section 5 presents a natural scaling principle for coarse-grid-resolution (160 km) simulations that retains the variability and coherent wave features of the moderate-grid simulations. Some discussion and concluding remarks are given in section 6.

## 2. Details of the model

In this section we briefly review the dynamical and physical features of the stochastic multicloud parameterization. The goal of this review is to highlight aspects of the model relevant to the discussion in the following sections and introduce modifications to the closures; thus, only the most salient features of the multicloud modeling framework are presented here. A more complete discussion of the stochastic model and multicloud framework is found in the original papers (KBM10; KM06a,b; KM08a,b).

### a. Dynamical core of the multicloud model

The multicloud parameterization framework assumes three heating profiles associated with the main cloud types that characterize organized tropical convective systems (Johnson et al. 1999): cumulus congestus clouds that heat the lower troposphere and cool the upper troposphere, through radiation and detrainment; deep convective towers that heat the whole tropospheric depth; and the associated lagging-stratiform anvils that heat the upper troposphere and cool the lower troposphere through evaporation of stratiform rain. Accordingly, Khouider and Majda (e.g., KM06a; KM08a) used the momentum and potential temperature equations for the first and second baroclinic modes of vertical structure, which are directly forced by deep convection and both congestus and stratiform clouds, respectively, as a minimal dynamical core that captures the main (linear response) effects of these three cloud types. Versions of this simple modeling framework that include effects of convective momentum transport (CMT) are found in Majda and Stechmann (2008, 2009) and Khouider et al. (2012). The multicloud model also carries equations for the vertically averaged moisture (water vapor mixing ratio), over the tropospheric depth, and bulk boundary layer dynamics averaged over the atmospheric boundary layer (ABL).

For convenience, the governing equations (along the equatorial ring, neglecting the effect of rotation) are reported in Tables 1 and 2 while the associated constants and parameters are reported in Table 3. All equations are given in the nondimensional form where the speed of the first baroclinic Kelvin waves, *c _{r}* ≈ 50 m s

^{−1}, is the velocity scale; the equatorial Rossby radius of deformation,

*L*≈ 1500 km, is the length scale;

_{e}*T*=

*L*/

_{e}*c*≈ 8.33 h is the time scale; and

_{r}Prognostic and diagnostic equations shared by the stochastic and deterministic models.

Convective heating closures for the stochastic and for the deterministic parameterizations.

Constants and parameters.

We note that the governing equations in Table 1 were derived under the tacit assumption that the three cloud types are associated with the following idealized heating profiles. Deep convection assumes a half-sine heating profile [sin(*πz*/*H _{T}*), where

*H*is the total height of the troposphere], while the stratiform and congestus heating profiles are based on a full sine [sin(2

_{T}*πz*/

*H*)]. While these heating profiles may deviate significantly from those observed in nature, especially for the congestus clouds whose upper-level radiative and evaporative cooling occur typically in a thin layer near the cloud top that does not exceed the freezing level, the resulting simple, two-baroclinic-mode model is very successful in representing convectively coupled equatorial waves and the MJO (KM06a; KM08a,b; Han and Khouider 2010; Khouider et al. 2011). Moreover, it is found in KM08a that the effects of asymmetry in the heating profiles do not induce any major changes in the qualitative behavior of the deterministic multicloud model or of the waves.

_{T}### b. The stochastic multicloud model

The stochastic multicloud parameterization is designed to capture the dynamical interactions among the three cloud types that characterize organized tropical convection and the environment using a coarse-grained lattice model (KBM10). To mimic the behavior within a typical GCM grid box, a rectangular *n* × *n* lattice is considered, where each element can be either occupied by a congestus, deep, or a stratiform cloud or is clear sky, through an order parameter that takes values of 0, 1, 2, or 3 on each lattice site. A continuous-time stochastic process is then defined by allowing the transitions, for individual cloud sites, from one state to another according to intuitive probability transition rates that depend on the large-scale resolved variables. These large-scale variables are the convective available potential energy integrated over the whole troposphere (CAPE), the convectively available energy integrated over the lower troposphere CAPE* _{l}* (see Table 2), and the dryness of the midtroposphere, which is a function of the difference between the atmospheric boundary layer (ABL) temperature

*θ*

_{eb}and the middle tropospheric potential temperature

*R*

_{ik}and the associated time scales

*τ*

_{ik}stated in Table 4 as well as function Γ defined below:

_{0}and

*T*

_{0}whose values are specified in Table 3.

Transition rates and time scales in the stochastic parameterization.

Notice that the assumption that the transition rates depend on the large-scale variables accounts for the feedback of the large scales on the stochastic model, while ignoring the interactions between the lattice sites all together implies that the stochastic processes associated with the different sites are identical. The latter simplification makes it easy to derive the stochastic dynamics for the GCM gridbox cloud coverages alone, which can be evolved without the detailed knowledge of the microstate configuration, by using a coarse-graining technique (Katsoulakis et al. 2003b; Khouider et al. 2003) that yields here a system of three birth–death-like processes, corresponding to the three cloud types. The resulting birth–death Markov system is easily evolved in time using Gillespie’s exact algorithm (Gillespie 1975, 1977). Thus, given the large-scale thermodynamic quantities, the stochastic process yields the dynamical evolution for the congestus, deep, and stratiform cloud fractions *σ _{c}*,

*σ*, and

_{d}*σ*, respectively.

_{s}From Eq. (2), we note that congestus heating *H _{c}* is proportional to the product of the congestus cloud fraction and square root of low-level CAPE. Thus both the congestus heating and congestus cloud fraction are tied to convectively available energy in the lower tropospheric region where congestus clouds are most active. Consistent with Lin and Neelin (2000) (see also Lin and Neelin 2003), the deep convective time scale

*τ*is inversely proportional to the stochastic area fraction of deep convection. At equilibrium, when

_{c}*τ*is set to the value

_{c}*θ*

_{eb}, the averaged atmospheric moisture

*q*, and the midtropospheric potential temperature

*θ*

_{1}+

*γ*

_{2}

*θ*

_{2}, through the coefficients

*a*

_{1},

*a*

_{2}, and

*a*

_{0}. The first two parameters satisfy (

*a*

_{1}+

*a*

_{2}= 1) and serve as a switch between CAPE and Betts–Miller parameterization, while the last parameter

*a*

_{0}parameterizes the convective response to fluctuations in the dry buoyancy (KM06a,b) Accordingly, the dry convective buoyancy parameter is set to a low value of

*a*

_{0}= 2 relative to the values previously used in KBM10 to enhance deep convection. A new feature of the model formulation is the stratiform heating closure [Eqs. (5) and (6)], which takes the same form as deep convective closure. This new formulation combined with fixed transition rates for the formation (from deep convective clouds) and decay of stratiform clouds simplifies the model’s complexity.

*H*+

_{d}*ξ*+

_{c}H_{c}*ξ*, where

_{s}H_{s}*ξ*and

_{c}*ξ*measure relative contributions of congestus and stratiform precipitation, respectively. Since the amount of congestus rain in the tropical ocean is on the order of 10% (Schumacher and Houze 2003), the parameter

_{s}*ξ*is set to zero. At radiative–convective equilibrium (RCE), the total precipitation heating is balanced by the imposed radiative cooling of 1 K day

_{c}^{−1}. Because the stratiform creation and dissipation rate is fixed relative to the deep convective cloud cover, it is easy to derive a closed expression for averaged stratiform rain fraction

*f*at RCE by invoking the mean value theorem (as shown in KM08a):

_{s}*ξ*is fixed so that the model yields 40% stratiform rain in accordance with observations (Schumacher and Houze 2003); that is, by setting

_{s}*f*= 0.4 in Eq. (7).

_{s}Aside from the changes in closures noted above, a small number of changes in tuning parameters (compared to KBM10), mainly aimed at improving the intermittency of the stochastic parameterization, will be considered in section 3 where the intermittent single column solutions are discussed and compared to previous results. These tuning parameter changes include CAPE and dryness adjustment parameters, the number of convective elements within one large-scale grid box, and certain changes to the cloud transition time scales.

### c. A paradigm model deterministic GCM convective parameterization with clear deficiencies

The design principles of the deterministic multicloud parameterization are based on the interactions of the same three cloud types. Its dynamical features and its capabilities, in terms of the impact of organized convection on the large-scale tropical circulation and convectively coupled waves, are demonstrated in Khouider and Majda (KM06a,b; KM07; KM08a,b) using both linear analysis and nonlinear simulations. The deterministic closure takes into account the energy available for congestus and deep convection (respectively

The key difference between this simulation and the well-tuned simulations of KM08a,b lies in two parameter changes. First, the stratiform rain fraction is increased from 0.1 to 0.4 in accordance with observations (Schumacher and Houze 2003). The higher stratiform rain fraction increases the stability of the waves in the simulation and results in reduced variability. Second, we use a relatively lower convective buoyancy parameter *a*_{0} to increases strength of the instability according to the parameter sensitivity tests reported in KM06a and KM08a; through *a*_{0}, positive *θ*_{1} anomalies constitute a negative feedback for deep convection, according to the deep convection closure formula in Table 2. Combined, the two parameter changes produce simulations where the variability is reduced to convectively coupled waves of the type shown in Fig. 1. These extremely stable waves propagate over the length of the domain at 22 m s^{−1} in both homogeneous SST and warm pool scenarios (KM08a; KM06b). A closer look at the structure of the waves reveals a convectively coupled wave with sharp deep convective peak and surprisingly little congestus heating. In fact, because of the low dry convective buoyancy frequency, unimpeded deep convection dominates congestus heating. The lack of congestus heating leads to vanishing of the heating field tilt that can be observed in nature and the deterministic multicloud model in an optimal parameter regime.

The time-average flow shown in Fig. 2, for a simulation with a nonuniform SST, in the form of a warm pool following KM08a,b [see also Eq. (8) in section 3b], reveals another undesirable feature of the detuned deterministic convective parameterization. The main feature of the Walker circulation is a sharp peak in the congestus and deep heating near the center of the warm pool. The sharp transition is characteristic of deterministic parameterization in poor parameter regimes. As seen in Fig. 3, convectively coupled waves travel the length of the domain and interact only weakly with the warm pool.

As in Fig. 1, but for (rows 1–3) the zonal structure of the time-averaged flow variables and for the horizontal–vertical structure of the time-averaged mean flow superimposed on (row 5) temperature anomalies and (row 6) the heating on a nonuniform SST background, mimicking the Indian Ocean western Pacific warm pool. The average is taken on the last 100 days of simulation (see KM07). Note that a dominant feature of the structure is a sharp peak in convection in the center of the warm pool.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 1, but for (rows 1–3) the zonal structure of the time-averaged flow variables and for the horizontal–vertical structure of the time-averaged mean flow superimposed on (row 5) temperature anomalies and (row 6) the heating on a nonuniform SST background, mimicking the Indian Ocean western Pacific warm pool. The average is taken on the last 100 days of simulation (see KM07). Note that a dominant feature of the structure is a sharp peak in convection in the center of the warm pool.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 1, but for (rows 1–3) the zonal structure of the time-averaged flow variables and for the horizontal–vertical structure of the time-averaged mean flow superimposed on (row 5) temperature anomalies and (row 6) the heating on a nonuniform SST background, mimicking the Indian Ocean western Pacific warm pool. The average is taken on the last 100 days of simulation (see KM07). Note that a dominant feature of the structure is a sharp peak in convection in the center of the warm pool.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

The *x–t* contours of the deviations from the time mean in Fig. 2 depicting streaks corresponding to convectively coupled waves circling the periodic domain that are only weakly interacting with the mean flow. Notice that congestus heating is active only within the warm pool region (in the middle of the domain) and is relatively weak.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

The *x–t* contours of the deviations from the time mean in Fig. 2 depicting streaks corresponding to convectively coupled waves circling the periodic domain that are only weakly interacting with the mean flow. Notice that congestus heating is active only within the warm pool region (in the middle of the domain) and is relatively weak.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

The *x–t* contours of the deviations from the time mean in Fig. 2 depicting streaks corresponding to convectively coupled waves circling the periodic domain that are only weakly interacting with the mean flow. Notice that congestus heating is active only within the warm pool region (in the middle of the domain) and is relatively weak.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

While the model produces nontrivial amount of variability, the structure of both the mean circulation and waves is not physical. Overall this illustrates one of the key weaknesses of conventional deterministic parameterizations used in operational GCMs. In this suboptimal regime, the model produces unphysical Walker circulation with weak variability about the mean. Furthermore, the variability in the simulation comes from extremely stable wave forms with unnatural structure. All of the above shortcomings will be corrected by the stochastic parameterization considered in the following sections.

## 3. Column model validation

In this section, we present some new results of the stochastic multicloud model in the context of single column simulations (KBM10). As such, the section is used to explain typical behavior of the stochastic multicloud parameterization in the absence of spatial variations and clarify the changes in some basic tuning parameters and time scales compared to KBM10. The second goal of the section is to study the sensitivity of the single column model to SST variations, which will be contrasted against SST gradient simulations with full-scale spatial effects presented in the next section.

The single column equations are obtained by disregarding spatial dependence components and the zonal wind in Table 1. As in KBM10, we employ a third-order Adams–Bashforth method to integrate the resulting ODE system. The coarse grained birth–death process is evolved in time by means of an acceptance–rejection Markov chain Monte Carlo method based on Gillespie’s exact algorithm (Gillespie 1975, 1977). The construction of such an algorithm as well as the associated reduction in computational cost is discussed thoroughly in KBM10.

### a. Single column simulations

A time series of a typical solution for the parameter regime described in Table 3 is plotted in Fig. 4. The top two panels present the plots of the large-scale variables *θ*_{1}, *θ*_{2}, *θ*_{eb}, and *q*. The middle two plots show the cloud fractions and associated heating profiles for the three cloud types while the bottom two panels show the total precipitation and large-scale quantities that drive the stochastic dynamics, namely *C _{l}*,

*C*, and

*D*(Table 3). The most notable feature is the time synchronization of the oscillations of the stochastic and deterministic variables, which leads to time series with frequent precipitation peaks of 10 K day

^{−1}and more intermittent large precipitation events on the order of 25 K day

^{−1}. For both the cloud fractions and resulting heating profiles, congestus bursts are followed by deep convective bursts, which in turn lead to stratiform peaks, consistent with the physical intuition utilized to design the model. The fluctuations in the cloud area fractions are directly related to changes in the large-scale fields. The increase in congestus area fraction is a direct response to low-level CAPE buildup (corresponding to the

*θ*

_{eb}peaks). Congestus heating anomalies then yield a rise in

*θ*

_{2}by direct heating, which yields a rise in

*θ*

_{em}. The moist atmosphere combined with high abundance of CAPE (which is not as sensitive to the second baroclinic mode heating as low-level CAPE) triggers the peak in

*σ*and associated deep convective heating. This is followed by trailing stratiform clouds, since there is a nonzero probability that a significant fraction of the deep convective clouds are converted into stratiform clouds.

_{d}Time evolution of a single column simulation showing (rows 1–2) the large-scale prognostic variables, (row 3) the (stochastic) cloud area fractions, (row 4) heating fields, and (row 5) normalized CAPE, CAPE* _{l}*, and dryness. The parameters assume the standard values in Tables 3 and 4.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Time evolution of a single column simulation showing (rows 1–2) the large-scale prognostic variables, (row 3) the (stochastic) cloud area fractions, (row 4) heating fields, and (row 5) normalized CAPE, CAPE* _{l}*, and dryness. The parameters assume the standard values in Tables 3 and 4.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Time evolution of a single column simulation showing (rows 1–2) the large-scale prognostic variables, (row 3) the (stochastic) cloud area fractions, (row 4) heating fields, and (row 5) normalized CAPE, CAPE* _{l}*, and dryness. The parameters assume the standard values in Tables 3 and 4.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

The main difference between the single column results reported here and the original KBM10 publication is the highly intermittent nature of the large precipitation events. The medium-strength convective events are frequent but not periodic, while the larger convective events are even more intermittent with standard deviation of the passage time equal to the mean passage time (indicative of Poisson distribution in time). The relationship between medium and large precipitation events is reminiscent of progressive deepening of the convection on multiple scales (Mapes et al. 2006). Small congestus cloud peaks are followed by small deep convective precipitation, driven primarily by congestus to deep conversion (Fig. 5). Deep convection is in turn followed by stratiform anvils, which cool and dry the boundary layer by downdrafts, thus reducing CAPE, and at the same time help moisten the lower troposphere for the next convective episode, via the evaporation of stratiform rain. These small-scale convective events are followed by larger congestus events, which moisten the environment to allow direct transition from clear sky to deep convective clouds, which account for a large percentage of the precipitation in the intermittent large-scale convective events.

As in Fig. 4, but for a short period of 2 days.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 4, but for a short period of 2 days.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 4, but for a short period of 2 days.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

This intermittence is due in part to the systematic choice of the tuning parameters and transition time scales motivated in part by the sensitivity studies in KBM10. The first factor contributing to the variability is the relatively low number of convective elements set to *N* = 30 × 30. It is easy to see that as the number of elements increases, the dynamics of the stochastic process converge to the behavior of deterministic mean field equations, thereby losing some of the inherent intermittency of the stochastic process. Second, the choice of the CAPE_{0} value, which can be viewed as an “activation energy,” is motivated by the observation that while large values of the threshold lead to large fluctuations in both the cloud area fractions and the large-scale climate variables, small values of the activation energy lead to intermittent large events on top of very weak and fast oscillations. The value of CAPE_{0} = 200 J kg^{−1} strikes a balance between the values originally considered in KBM10. Meanwhile, the value of parameter *T*_{0} controlling the influence of the dryness in the stochastic model is increased to yield more dramatic dependence on the moisture field.

The transition time scales (provided in Table 4) also play a critical role in the intermittency of the solution. A crucial design feature of the present parameter regime is the parity between congestus birth and decay time scales. While previous studies of organized tropical convection (e.g., Johnson et al. 1999; Mapes 2000; Khouider and Majda 2006a,b, etc.) suggest that cloud decay time scales are generally longer than cloud creation time scales, we choose to relax this proposition for congestus clouds. In this particular model, fast decay of the congestus clouds can be physically justified as a proxy for detrainment mechanism observed in nature (but otherwise missing in the model). While individually neither of the above parameters leads to dramatic improvement in variability, all the parameter changes combined with closure modifications (formulated in the previous section) break the stochastic resonance observed in the KBM10 and yield the highly intermittent solution described above.

### b. Sensitivity of single column simulations to SST variation: A mock warm pool

*x*and impose artificially

*x*,

*t*) simulations of the following section, we choose to name this simulation a “mock” warm pool because of the absence of spatial dependency in the uncoupled column model.

The top two panels in Fig. 6 present the plots of the large-scale variables *θ*_{1}, *θ*_{2}, *θ*_{eb}, and *q*, while the next two plots show the cloud fractions and associated heating profiles as functions of the variable *x*. Each point represents the 50-day average of the single column simulation with corresponding SST perturbation (plotted in the bottom panel of the figure). It is clear that an increase in ABL equivalent potential temperature in the “center” of this mock warm pool leads to increased congestus activity, which in turn leads to abundance of deep and consequently stratiform convection in the high SST region. The increase in convection leads to heating of the first and second baroclinic mode. Deep convective heating is more intermittent in low SST areas and on average lower but more persistent in high SST area. Meanwhile, the cloud fractions for all three cloud types peak at the center of the mock warm pool. The lack of the pronounced deep convective heating peak in the center points to the importance of the spatial effects, most importantly moisture convergence. While the experiment does not produce a realistic model for the Walker climatology, the parameterization is not overly sensitive to SST variation (convection in various amounts exists for all perturbed states). In fact, this study is an analog of single column analysis for the homogeneous SST simulations and can be used effectively to benchmark the stochastic multicloud model in warm pool type simulations as illustrated above. While the results are plotted in a manner that facilitates direct comparison to the (*x*, *t*) simulations, and thus allows isolation of the purely spatial effects, these results should not be confused with the actual Walker cell simulations in sections 4 and 5.

A mock Walker circulation formed by the averaged response of single column simulations for a sequence of SST values mimicking a warm pool. (bottom) The perturbation profile is identical to the one used for the full *x–t* simulation. Parameters are as in Tables 3 and 4 and *x* = 0 to *x* = 40 000 km according to Eq. (8).

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

A mock Walker circulation formed by the averaged response of single column simulations for a sequence of SST values mimicking a warm pool. (bottom) The perturbation profile is identical to the one used for the full *x–t* simulation. Parameters are as in Tables 3 and 4 and *x* = 0 to *x* = 40 000 km according to Eq. (8).

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

A mock Walker circulation formed by the averaged response of single column simulations for a sequence of SST values mimicking a warm pool. (bottom) The perturbation profile is identical to the one used for the full *x–t* simulation. Parameters are as in Tables 3 and 4 and *x* = 0 to *x* = 40 000 km according to Eq. (8).

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

## 4. Stochastic parameterization in a moderate-resolution toy GCM

We solve the equations described in Tables 1, 2, and 3 in (*x*, *t*) variables on a 40 000-km periodic ring representing the perimeter of the earth at the equator. We use an operator time-splitting strategy where the conservative terms are discretized and solved by a nonoscillatory central scheme while the remaining convective forcing terms are handled by a second-order Runge–Kutta method (Khouider and Majda 2005a,b). The stochastic component of the scheme is resolved using Gillespie’s exact algorithm described in the previous section. The coupled model is integrated for a total period of 1250 days with a mesh size of 40 km and a time step of 2 min. The first 250 days are discarded as a transient period, while the last 1000 days are used for the calculation of the statistics. The last 30 days of the simulation are used to display the features of the solution.

### a. Homogenous SST background: Aquaplanet

The simulation for a homogeneous SST background is presented in Figs. 7 and 8. The first figure shows the velocity and temperature modes, while the second figure shows the moisture and ABL equivalent temperature followed by deep and congestus heating, as well as deep and congestus cloud fractions. Considering the contours of deep and congestus heating we note some of the prominent features of the simulation: synoptic and mesoscale convective systems, large intermittent congestus cloud decks, convectively coupled waves, and small-scale convective events. As was the case with the single column results, the boundary layer temperature is the main source of the variability. The top-right panel of Fig. 8 shows the ABL equivalent potential temperature, which correlates with the large peaks of congestus heating. These large congestus cloud decks moisten the lower troposphere and create favorable environment for large deep convective peaks. Radiating away from the near-stationary decks are convectively coupled waves which move with an average speed of 17 m s^{−1}. Smaller-scale structures such as intermittent congestus and deep convective events can be seen throughout the domain, consistent with observations.

Plot of *x*–*t* contours of (top left) first and (top right) second baroclinic velocity. (bottom) As in top, but for potential temperature components over the last 30 days of simulation. Stochastic model with spatial dependence on a 40-km grid, with homogeneous SST background.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Plot of *x*–*t* contours of (top left) first and (top right) second baroclinic velocity. (bottom) As in top, but for potential temperature components over the last 30 days of simulation. Stochastic model with spatial dependence on a 40-km grid, with homogeneous SST background.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Plot of *x*–*t* contours of (top left) first and (top right) second baroclinic velocity. (bottom) As in top, but for potential temperature components over the last 30 days of simulation. Stochastic model with spatial dependence on a 40-km grid, with homogeneous SST background.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 7, but for (top) moisture and ABL equivalent potential temperature, (middle) deep and congestus convective heating fields, and (bottom) (stochastic) cloud fractions.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 7, but for (top) moisture and ABL equivalent potential temperature, (middle) deep and congestus convective heating fields, and (bottom) (stochastic) cloud fractions.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 7, but for (top) moisture and ABL equivalent potential temperature, (middle) deep and congestus convective heating fields, and (bottom) (stochastic) cloud fractions.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

The congestus cloud decks appear intermittently in space and in time and remain stationary while generating gravity and convectively coupled waves. While the average lifespan of the convectively coupled waves seems to be on the order of a week, the waves travel far enough to interact with convectively coupled waves from other cloud decks. As is the case with the single column results, the homogeneous SST simulations show a variety of the convective events of different scales and magnitudes. While large congestus cloud decks give birth to large precipitation events and the convectively coupled waves, the convectively coupled waves and the deep convective events themselves carry smaller amounts of congestus cloud cover. The structure of the waves is considered in the next subsection.

### b. Warm pool

Now we impose a nonuniform SST background climate so that

First, we show the zonal structure of the resulting mean circulation in Fig. 9. The simulation results in a nontrivial Walker-like circulation with 10 m s^{−1} maximum horizontal velocity and 2 cm s^{−1} maximum updraft. The high SST creates an enhanced convection region inside of the warm pool, resulting in a peak of deep convection in the center of the warm pool that drives the circulation; meanwhile, on the edges of the warm pool the pronounced peaks in deep convection are due to intermittent convectively coupled waves radiating away from the warm pool (going through a mature stage and then losing strength and moisture as they move away from the warm pool). It is important to recall that the mock warm pool simulation, which tracks the response of single column to SST variation, lacks the three-peak deep convective maximum structure. We claim that enhanced deep convective heating inside the warm pool is due to correlated spatial effects, such as moisture convergence, as is the case for peaks on the boundary of the warm pool associated with transient convectively coupled waves. Small-scale deep convective heating exists in the low SST zone; it is fueled by low first baroclinic potential temperature anomalies (high relative humidity). It is worth mentioning that the stochastic model avoids the sharp peaks characteristic of the deterministic model in the suboptimal regime; the stochastic model allows nontrivial interactions between the waves and the warm pool forcing. The vertical structure of the resulting simulation can rival the ones produced by high-resolution CRMs. For example, Grabowski et al. (2000) report circulation with a somewhat weaker horizontal velocity (8 m s^{−1}) and a stronger updraft (4 cm s^{−1}) as results of an SST gradient simulation in CRM runs on smaller 4000-km domain with similar vertical structure. The variability in the present stochastic simulation qualitatively resembles that in these vastly more expensive CRM simulations.

Zonal structure of the time-averaged flow variables and vertical structure of the time-averaged mean flow for the stochastic model: the average is taken on the last 1000 days of simulation. Nonuniform SST background mimics a warm pool. The parameters are as in Tables 3 and 4 and

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Zonal structure of the time-averaged flow variables and vertical structure of the time-averaged mean flow for the stochastic model: the average is taken on the last 1000 days of simulation. Nonuniform SST background mimics a warm pool. The parameters are as in Tables 3 and 4 and

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Zonal structure of the time-averaged flow variables and vertical structure of the time-averaged mean flow for the stochastic model: the average is taken on the last 1000 days of simulation. Nonuniform SST background mimics a warm pool. The parameters are as in Tables 3 and 4 and

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

The structure of the deviations from the mean circulation (i.e., the waves) is plotted in Figs. 10 and 11 following the same format as Figs. 7 and 8. The main driver of the circulation is the presence of large congestus cloud decks and associated large-scale deep precipitation events. While in the homogeneous background simulation these cloud decks are intermittently created throughout the domain, the imposed SST gradient leads to the localization of the large congestus decks inside the high SST region. As in the homogenous background scenario, convectively coupled waves radiate away from the large congestus cloud decks that are now confined to the warm pool. These structures are short lived but create drier waves that propagate farther away. The resulting precipitation profiles again resemble CRM simulations of Grabowski et al. (2000), where convectively coupled wave organization is observed in the high SST area. However, unlike Grabowski et al. (2000), intermittent deep convection episodes are also observed away from the warm pool region, perhaps due to our much larger spatial domain.

Plot of *x–t* contours of deviations from the time mean in Fig. 9 of (left) first and (right) second baroclinic components of (top) velocity and (bottom) potential temperature over the last 30 days of simulation on 40-km resolution grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Plot of *x–t* contours of deviations from the time mean in Fig. 9 of (left) first and (right) second baroclinic components of (top) velocity and (bottom) potential temperature over the last 30 days of simulation on 40-km resolution grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Plot of *x–t* contours of deviations from the time mean in Fig. 9 of (left) first and (right) second baroclinic components of (top) velocity and (bottom) potential temperature over the last 30 days of simulation on 40-km resolution grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 10, but for (top) moisture and ABL equivalent potential temperature, (middle) deep and congestus convective heating fields, and (bottom) (stochastic) cloud fractions, over the last 30 days of simulation on 40-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 10, but for (top) moisture and ABL equivalent potential temperature, (middle) deep and congestus convective heating fields, and (bottom) (stochastic) cloud fractions, over the last 30 days of simulation on 40-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 10, but for (top) moisture and ABL equivalent potential temperature, (middle) deep and congestus convective heating fields, and (bottom) (stochastic) cloud fractions, over the last 30 days of simulation on 40-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

A variety of convectively coupled and dry waves is present in warm pool and homogeneous SST simulations. However, the most interesting physical structures in the simulation are the CCWs that move away from the warm pool at speed of approximately 17 m s^{−1}. The structure of these waves is illustrated in Fig. 12, the waves are averaged in the moving reference frame resulting in maximum horizontal velocity of 3 m s^{−1} and 3 cm s^{−1} updraft. As was the case in the single column simulation, both cloud fractions and heating field exhibit congestus to deep to stratiform pattern consistent with the design principles of the multicloud model. Furthermore, the snapshots of the evolution of the wave reveal that the initial stage of the wave is dominated by congestus clouds near the warm pool. As the wave moves away from the warm pool area, deep convection peaks around 5000 km away from the center of the warm pool. The (*x*, *z*) structure of the heating field reveals characteristic tilt observed in nature (Wheeler and Kiladis 1999; Kiladis et al. 2005) and consistent with the progressive deepening of convection from shallow to congestus to deep to stratiform seen in detailed small-domain CRM simulations (Waite and Khouider 2009).

Wave composite of the typical convectively coupled wave for the stochastic model with a warm pool SST. (rows 1–4) The structure of the anomalies of indicated variables. (rows 6–7) The zonal vertical velocity arrows overlaid on top of the (total) potential temperature anomalies and total heating perturbation, respectively. The composite is obtained by averaging one particular wave signal from Figs. 10 and 11, over the lifespan of the corresponding wave, following a reference frame moving westward (with the wave) at 17 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Wave composite of the typical convectively coupled wave for the stochastic model with a warm pool SST. (rows 1–4) The structure of the anomalies of indicated variables. (rows 6–7) The zonal vertical velocity arrows overlaid on top of the (total) potential temperature anomalies and total heating perturbation, respectively. The composite is obtained by averaging one particular wave signal from Figs. 10 and 11, over the lifespan of the corresponding wave, following a reference frame moving westward (with the wave) at 17 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Wave composite of the typical convectively coupled wave for the stochastic model with a warm pool SST. (rows 1–4) The structure of the anomalies of indicated variables. (rows 6–7) The zonal vertical velocity arrows overlaid on top of the (total) potential temperature anomalies and total heating perturbation, respectively. The composite is obtained by averaging one particular wave signal from Figs. 10 and 11, over the lifespan of the corresponding wave, following a reference frame moving westward (with the wave) at 17 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

Compared to the reference deterministic parameterization, the stochastic model avoids the oversensitivity to SST variation and reproduces a realistic Walker circulation (without the extremely sharp peak in deep convection), which is qualitatively similar to the computationally expensive CRM results of Grabowski et al. (2000). The structure of the convectively coupled waves is greatly improved as well as the interactions between the mean circulations and mesoscale convective systems.

## 5. Stochastic parameterization in a coarse-resolution GCM

This section considers scalability of the stochastic multicloud parameterization to the coarse GCM resolution (160 km). We propose an intuitive time scale dilation, which preserves the variability and statistical structure of coherent features described in the medium-resolution study of the previous section. The qualitative analysis of the variability of the model is done in the warm pool setting, through judicious qualitative comparison of the scaled stochastic model to the deterministic GCM parameterization with clear deficiencies.

The model for equatorial flow can be considered as a chain of rectangular lattices. All the results described thus far have been achieved by considering rectangular lattices of size 40 km each occupied by *n* × *n* sites where *n* is set to 30. In this framework, an individual convective element has an area of 1.3^{2} km^{2} and corresponds to a transition time scale on the order of hours (as reported in Table 4). When the model is applied to a coarser resolution of 160 km, it is natural to consider modifications to the number of convective elements and their corresponding transition time scales. Naturally, the number of convective elements can be increased to *n* = 120, preserving the area to time scale ratio for all convective elements. This, however, is a step toward the mean field limit (KBM10) and naturally leads to almost deterministic dynamics with reduced intermittency. Alternatively, we many consider the role of time scale adjustment parameter *τ*_{grid} that is set to one in the previous medium-resolution simulation. Keeping the number of elements the same, we consider making transition time scales longer by increasing in the value of *τ*_{grid}. The equilibrium distribution of convective elements is invariant under this transformation. Thus, keeping total number of convective elements the same, we simply associate larger convective sites with larger time scales, without disturbing the equilibrium structure of the solution. While it is not possible to use this argument to extend the model fully to a very coarse synoptic-scale grid, both 40-km and 160-km resolutions are well within the mesoscale framework of the proposed stretched building block hypothesis (Mapes et al. 2006). Alternatively, variation of the parameter *τ*_{grid} can be viewed as a systematic sensitivity study of the effects of the time scales on the coarse-resolution stochastic parameterization.

Motivated by the above discussion, we consider a set of numerical simulations in the warm pool environment. The strength of the mean circulation and standard deviation of deep convective and congestus heating fields are summarized in Table 5. As expected, an increase in the number of cloud sites, corresponding to a shift toward the mean field equations, results in a decrease in the variability of the solutions. While the mean field behavior of parameterization is interesting, the decrease in variability of the solution is counterproductive. Thus, for the remainder of the section we fix the number of cloud sites at *n* = 30 and experiment with the time scale adjustment only. The values of time scale dilation parameter *τ*_{grid} = 1, 3, 4, 5, 6, and 16 are considered. Notice that the case of *τ*_{grid} = 1 corresponds to a simple increase in the resolution of the stochastic lattice without any parameter changes and leads to a decrease of variability. Generally, the variance of congestus heating goes down as larger time scale dilations are introduced; meanwhile, the variance of deep convection peaks near *τ*_{grid} = 5. While the intuition suggests a spatial scaling of 16 (which preserves the area to time scale ratio), the simulations in this regime are characterized by a weak variability. Nonetheless, even this weakly fluctuating circulation of the stochastic model produces a variability comparable to the deterministic parameterization simulations on the coarse GCM grid, with greatly improved structure of mean circulation and convectively coupled waves.

Mean circulation strength and variability of heating fields for the stochastic and deterministic parameterizations under different scalings.

Overall, the solution with *τ*_{grid} = 4 has a balance of deep and congestus convection that produces the highest updraft of the simulation set. The mean of this circulation is plotted in Fig. 13. The heating structure, which resembles the moderate resolution simulation, is characterized by the trimodality of the convective heating field. As before, the deep convective heating peaks outside of the warm pool correspond to the mature stage of CCWs moving away from the large and intermittent congestus cloud decks found in the center of the warm pool. The main difference between the moderate- and coarse-resolution simulations is that the deep convective peak in the center is much higher in the coarse resolution. The increase in the maximum updraft of the coarse-resolution simulation correlates to the enhanced deep convective heating inside of the warm pool. As for the moderate grid counterpart, the coarse-grid stochastic parameterization avoids the nonphysical sharp peak seen in the deterministic parameterization. The deviations from the mean are plotted in Figs. 14 and 15. We note that while the complexity of the structures is somewhat reduced, the mean features of the moderate-resolution simulations are preserved. As before, large congestus cloud decks are confined to the warm pool area triggering convectively coupled waves that move away from the warm pool. The structure of such waves, given Fig. 16, qualitatively resembles its moderate-resolution counterpart (plotted in Fig. 12). Small-scale features associated with decaying CCWs as well as random convective episodes are seen outside of the warm pool.

As in Fig. 9, but for the 160-km-resolution grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 9, but for the 160-km-resolution grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 9, but for the 160-km-resolution grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 10, but for 160-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 10, but for 160-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 10, but for 160-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 11, but for 160-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 11, but for 160-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 11, but for 160-km grid.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 12, but for the 160-km resolution grid. Notice that the wave is moving eastward.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 12, but for the 160-km resolution grid. Notice that the wave is moving eastward.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

As in Fig. 12, but for the 160-km resolution grid. Notice that the wave is moving eastward.

Citation: Journal of the Atmospheric Sciences 69, 3; 10.1175/JAS-D-11-0148.1

## 6. Concluding summary and discussion

Here a modified version of the stochastic multicloud model is used to study flows above the equator without rotation effects. As in KBM10, the model is based on a coarse-grained Markov chain lattice model where each lattice site takes discrete values from 0 to 3 according to whether the site is clear sky or occupied by a congestus, deep, or stratiform cloud. The convective elements of the model interact with each other and with the large-scale environmental variables through CAPE and middle troposphere dryness. A few changes in the closure formulations are also introduced in order to improve the variability and the overall qualitative behavior of stochastic model. The major modifications to the original model include a direct dependence of congestus cloud cover on low-level CAPE, a simplified stratiform heating closure, and the inclusion of stratiform rain in the precipitation budget. Additionally, minor changes in tuning parameters and time scales are validated in single column simulations mimicking typical behavior of a GCM grid box. In this context, the combined effect of the changes yields highly intermittent solutions that capture the progressive deepening of tropical convection on multiple scales. In particular, small-scale convective events precondition the environment for large-scale precipitation. While both small and large convective events follow congestus to deep to stratiform patterns, the prominent events are characterized by larger deep convective heating relative to the congestus component (due to direct clear sky to deep convection transitions in the preconditioned environment).

The parameterization is used in an aquaplanet setting to study flows above the equator without rotation effects on a moderate (40 km)-resolution domain. Detailed simulations show a menagerie of intermittent synoptic and mesoscale convective systems as well as the emergence of 17 m s^{−1} convectively coupled waves originating from large congestus cloud decks. Adhering to the design principles of the model, these intermittent and unstable waves have the characteristic congestus to deep to stratiform heating pattern and tilted heating field observed in nature. The introduction of an SST gradient leads to a localization of the large congestus cloud decks within the high SST region, which in turn gives rise to large deep convective features that drive a more realistic Walker-type circulation. The convectively coupled waves propagate away from the boundaries of the congestus cloud decks resulting in high variability, which is further enhanced by intermittent small-scale convective features away from the warm pool area. Both the structure of the mean circulation and waves are comparable to the results of CRM simulations of Grabowski et al. (2000). The advantages of using the stochastic parameterization are particularly apparent when the results are compared to the suboptimal deterministic (conventional) parameterization in this paradigm setting.

The design principle from KBM10 allows for a natural transition time scale dilation that improves the performance of the model on a coarse GCM size mesh (160 km). A sensitivity study of the scaling parameter, in the SST gradient setting, shows that this principle can be successfully applied to produce coarse simulations that retain the variability and the statistical structure of coherent features of the medium-resolution solution. For both the medium and coarse resolutions, the variability of the stochastic parameterization exceeds the variability of the deterministic parameterization run over a medium-resolution grid. Furthermore, the variability comes from coherent structures as in nature. In the present idealized paradigm setting, the above results illustrate that the stochastic parameterization can successfully address the problem of missing tropical variability in determin-istic GCM parameterizations.

While the main goal of this paper is to show how “stochasticization” could improve a deficient (deterministic) convective parameterization, the comparison of the stochastic model with the well-tuned deterministic multicloud model could also be interesting. For example, a direct comparison between the present results and those obtained in KM08a shows that the structure of the stochastic waves, especially in terms of the contribution of the congestus heating, is more realistic (compare our Figs. 12 and 16 to Fig. 7 of KM08a). The increased congestus heating in the stochastic waves leads to a significant improvement of in the tilt of the overall heating profile of the waves.

It is worthwhile noting here that the main differences between the stochastic and deterministic parameterizations reside in 1) the fluctuations of the heating rates induced by the stochastic area fractions and 2) the systematic transitions between the various cloud types, essentially from congestus to deep and from deep to stratiform. All other parameters and closure formulations remained identical. In essence, we demonstrated here that this multicloud stochasticization framework can be easily exported to conventional cumulus parameterizations to improve the ability of operational GCMs to simulate the variability of organized tropical convection and convectively coupled waves. Such studies are currently being conducted by the authors in collaboration with other scientists and will be reported elsewhere in the near future.

## Acknowledgments

The research of B. K. is supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the Canadian Foundation for Climate and Atmospheric Sciences. The research of A. J. M. is supported by the National Science Foundation (NSF) Grant DMS-0456713, the office of Naval Research (NR) Grant N00014-05-1-0164, and the Defense Advanced Projects Agency Grant N0014-07-1-0750. Y. F. is a postdoctoral fellow supported through A.J.M.’s above NSF and ONR grants. This research is partly achieved at the Institute for Pure and Applied Mathematics when both Y. F. and B. K. were taking part as long-term visitors in the context of the long program on Climate and Data Hierarchies and A. J. M. was a main organizer and participant.

## REFERENCES

Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I.

,*J. Atmos. Sci.***31**, 674–701.Betts, A. K., and M. J. Miller, 1986: A new convective adjustment scheme. Part II: Single column tests using GATE wave, BOMEX, ATEX and arctic air-mass data sets.

,*Quart. J. Roy. Meteor. Soc.***112**, 693–709.Buizza, R., M. Miller, and T. N. Palmer, 1999: Stochastic representation of model uncertainties in the ECMWF Ensemble Prediction System.

,*Quart. J. Roy. Meteor. Soc.***125**, 2887–2908.Gillespie, D. T., 1975: An exact method for numerically simulating the stochastic coalescence process in a cloud.

,*J. Atmos. Sci.***32**, 1977–1989.Gillespie, D. T., 1977: Exact stochastic simulation of coupled chemical reactions.

,*J. Phys. Chem.***81**, 2340–2361.Grabowski, W. W., 2001: Coupling cloud processes with the large-scale dynamics using the cloud-resolving convection parameterization (CRCP).

,*J. Atmos. Sci.***58**, 978–997.Grabowski, W. W., 2003: MJO-like systems and moisture-convection feedback in idealized aquaplanet simulations.

*Proc. Workshop on Simulation and Prediction of Intraseasonal Variability with Emphasis on the MJO,*Reading, United Kingdom, CLIVAR/ECMWF, 67–72. [Available online at http://www.ecmwf.int/publications/library/ecpublications/_pdf/workshop/2003/MJO/ws_mjo_grabowski.pdf.]Grabowski, W. W., 2004: An improved framework for superparameterization.

,*J. Atmos. Sci.***61**, 1940–1952.Grabowski, W. W., and P. K. Smolarkiewicz, 1999: CRCP: A cloud resolving convection parameterization for modeling the tropical convecting atmosphere.

,*Physica D***133**, 171–178.Grabowski, W. W., J.-I. Yano, and M. W. Moncrieff, 2000: Cloud resolving modeling of tropical circulations driven by large-scale SST gradients.

,*J. Atmos. Sci.***57**, 2022–2040.Han, Y., and B. Khouider, 2010: Convectively coupled waves in a sheared environment.

,*J. Atmos. Sci.***67**, 2913–2942.Hendon, H. H., and B. Liebmann, 1994: Organization of convection within the Madden–Julian oscillation.

,*J. Geophys. Res.***99**, 8073–8084.Horenko, I., 2011: Nonstationarity in multifactor models of discrete jump processes, memory, and application to cloud modeling.

,*J. Atmos. Sci.***68**, 1493–1506.Johnson, R. H., T. M. Rickenbach, S. A. Rutledge, P. E. Ciesielski, and W. H. Schubert, 1999: Trimodal characteristics of tropical convection.

,*J. Climate***12**, 2397–2418.Kain, J. S., and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization.

,*J. Atmos. Sci.***47**, 2784–2802.Katsoulakis, M. A., A. J. Majda, and D. G. Vlachos, 2003a: Coarse-grained stochastic processes for microscopic lattice systems.

,*Proc. Natl. Acad. Sci. USA***100**, 782–787.Katsoulakis, M. A., A. J. Majda, and D. G. Vlachos, 2003b: Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems.

,*J. Comput. Phys.***186**, 250–278.Khouider B, and A. J. Majda, 2005a: A non oscillatory balanced scheme for an idealized tropical climate model; Part I: Algorithm and validation.

,*Theor. Comput. Fluid Dyn.***19**, 331–354.Khouider B, and A. J. Majda, 2005b: A non oscillatory balanced scheme for an idealized tropical climate model; Part II: Nonlinear coupling and moisture effects.

,*Theor. Comput Fluid Dyn.***19**, 355–375.Khouider B, and A. J. Majda, 2006a: Multicloud convective parametrizations with crude vertical structure.

,*Theor. Comput. Fluid Dyn.***20**, 351–375.Khouider B, and A. J. Majda, 2006b: A simple multicloud parameterization for convectively coupled tropical waves. Part I: Linear analysis.

,*J. Atmos. Sci.***63**, 1308–1323.Khouider B, and A. J. Majda, 2007: A simple multicloud parameterization for convectively coupled tropical waves. Part II: Nonlinear simulations.

,*J. Atmos. Sci.***64**, 381–400.Khouider B, and A. J. Majda, 2008a: Equatorial convectively coupled waves in a simple multicloud model.

,*J. Atmos. Sci.***65**, 3376–3397.Khouider B, and A. J. Majda, 2008b: Multicloud models for organized tropical convection: Enhanced congestus heating.

,*J. Atmos. Sci.***65**, 895–914.Khouider B, A. J. Majda, and M. A. Katsoulakis, 2003: Coarse-grained stochastic models for tropical convection and climate.

,*Proc. Natl. Acad. Sci. USA***100**, 11 941–11 946.Khouider B, J. Biello, and A. Majda, 2010: A stochastic multicloud model for tropical convection.

,*Commun. Math. Sci.***8**, 187–216.Khouider B, A. St-Cyr, A. J. Majda, and J. Tribbia, 2011: The MJO and convectively coupled waves in a coarse-resolution GCM with a simple multicloud parameterization.

,*J. Atmos. Sci.***68**, 240–264.Khouider B, Y. Han, and J. Biello, 2012: Convective momentum transport in a simple multicloud model.

,*J. Atmos. Sci.***69**, 281–302.Kiladis, G. N., K. H. Straub, and P. T. Haertel, 2005: Zonal and vertical structure of the Madden–Julian oscillation.

,*J. Atmos. Sci.***62**, 2790–2809.Kuo, H. L., 1974: Further studies of the parameterization of the influence of cumulus convection on large-scale flow.

,*J. Atmos. Sci.***31**, 1232–1240.Lau, W. K. M., and D. E. Waliser, 2005:

*Intraseasonal Variability in the Atmosphere–Ocean Climate System*. Springer-Verlag, 436 pp.Lin, J.-L., and Coauthors, 2006: Tropical intraseasonal variability in 14 IPCC AR4 climate models. Part I: Convective signals.

,*J. Climate***19**, 2665–2690.Lin, J. W.-B., and J. D. Neelin, 2000: Influence of a stochastic moist convective parameterization on tropical climate variability.

,*Geophys. Res. Lett.***27**, 3691–3694.Lin, J. W.-B., and J. D. Neelin, 2003: Toward stochastic deep convective parameterization in general circulation models.

,*Geophys. Res. Lett.***30**, 1162, doi:10.1029/2002GL016203.Majda, A. J., 2007: Multiscale models with moisture and systematic strategies for superparameterization.

,*J. Atmos. Sci.***64**, 2726–2734.Majda, A. J., and B. Khouider, 2002: Stochastic and mesoscopic models for tropical convection.

,*Proc. Natl. Acad. Sci. USA***99**, 1123–1128.Majda, A. J., and S. N. Stechmann, 2008: Stochastic models for convective momentum transport.

,*Proc. Natl. Acad. Sci. USA***105**, 17 614–17 619.Majda, A. J., and S. N. Stechmann, 2009: A simple dynamical model with features of convective momentum transport.

,*J. Atmos. Sci.***66**, 373–392.Majda, A. J., C. Franzke, and B. Khouider, 2008: An applied mathematics perspective on stochastic modelling for climate.

,*Philos. Trans. Roy. Soc.***366A**, 2427–2453.Manabe, S., J. Smagorinsky, and R. F. Strickler, 1965: Simulated climatology of a general circulation model with a hydrologic cycle 1.

,*Mon. Wea. Rev.***93**, 769–798.Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model.

,*J. Atmos. Sci.***57**, 1515–1535.Mapes, B. E., S. Tulich, J. Lin, and P. Zuidema, 2006: The mesoscale convection life cycle: Building block or prototype for large-scale tropical waves?

,*Dyn. Atmos. Oceans***42**, 3–29.Moncrieff, M., and E. Klinker, 1997: Organized convective systems in the tropical western Pacific as a process in general circulation models: A TOGA COARE case-study.

,*Quart. J. Roy. Meteor. Soc.***123**, 805–827.Moncrieff, M., M. Shapiro, J. Slingo, and F. Molteni, 2007: Collaborative research at the intersection of weather and climate.

,*WMO Bull.***56**, 204–211.Nakazawa, T., 1974: Tropical super clusters within intraseasonal variation over the western Pacific.

,*J. Meteor. Soc. Japan***66**, 823–839.Palmer, T. N., 2001: A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models.

,*Quart. J. Roy. Meteor. Soc.***127**, 279–304.Randall, D., M. Khairoutdinov, A. Arakawa, and W. Grabowski, 2003: Breaking the cloud parameterization deadlock.

,*Bull. Amer. Meteor. Soc.***84**, 1547–1564.Schumacher, C., and R. Houze, 2003: Stratiform rain in the tropics as seen by the TRMM precipitation radar.

,*J. Climate***16**, 1739–1756.Scinocca, J. F., and N. A. McFarlane, 2004: The variability of modeled tropical precipitation.

,*J. Atmos. Sci.***61**, 1993–2015.Slingo, J. M., and Coauthors, 1996: Intraseasonal oscillations in 15 atmospheric general circulation models: Results from an AMIP diagnostic subproject.

,*Climate Dyn.***12**, 325–357.Stechmann, S. N., and J. D. Neelin, 2011: A stochastic model for the transition to strong convection.

, 68, 2955–2970.*J. Atmos. Sci.*Takayabu, Y. N., 1994: Large-scale cloud disturbances associated with equatorial waves. Part I: Spectral features of the cloud disturbances.

,*J. Meteor. Soc. Japan***72**, 433–448.Waite, M. L., and B. Khouider, 2009: Boundary layer dynamics in a simple model for convectively coupled gravity waves.

,*J. Atmos. Sci.***66**, 2780–2795.Wheeler, M., and G. N. Kiladis, 1999: Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber–frequency domain.

,*J. Atmos. Sci.***56**, 374–399.Xing, Y., A. J. Majda, and W. W. Grabowski, 2009: New efficient sparse space–time algorithms for superparameterization on mesoscales.

,*Mon. Wea. Rev.***137**, 4307–4324.Zhang, C., 2005: Madden–Julian oscillation.

,*Rev. Geophys.***43**, RG2003, doi:10.1029/2004RG000158.Zhang, G. J., and N. A. McFarlane, 1995: Role of convective-scale momentum transport in climate simulation.

,*J. Geophys. Res.***100**, 1417–1426.