## 1. Introduction

The interplay between moist convection and the large-scale flow is the fundamental feature of the tropical atmosphere. However, the extreme range of spatial and temporal scales involved makes it difficult to resolve all relevant processes in numerical models. In large-scale models, this issue has traditionally been addressed through the use of convective parameterizations that account for effects of convective motions on the mean atmospheric temperature and humidity profiles. It is well recognized, however, that convective parameterizations fail to reproduce many important features of the tropical atmosphere. This is partly because many aspects of convection, such as downdrafts, cold pools, and mesoscale organization, are either excluded or poorly represented in the parameterizations. Moreover, the parameterizations often do not reproduce the intrinsic intermittency of moist convection. This motivates the development of new approaches to improve the representation of convection in multiscale simulations of the tropical atmosphere.

One way to improve such simulations is to take advantage of cloud-resolving modeling. Cloud models emerged in the 1970s (e.g., Steiner 1973; Schlesinger 1975; Klemp and Wilhelmson 1978; Clark 1979) to study individual clouds in short simulations (tens of minutes) and typically applied idealized forcing techniques (e.g., initiating cloud development via a warm bubble). More recently such models have been used in significantly longer simulations (days and weeks) and apply large computational domains. Such simulations are often driven by observationally based time-evolving large-scale forcings and thus allow for better comparison with observations (e.g., Grabowski et al. 1996, 1998; Xu and Randall 1996; Xu et al. 2002; Fridlind et al. 2012, among many others).

Cloud-resolving models solve nonhydrostatic governing equations and allow convective development in conditionally unstable conditions. The horizontal resolution of ~1 km is high enough for the simulation of the dynamical evolution of individual clouds, with microphysical, turbulent, and radiative processes required to be parameterized. Explicit representation of cloud dynamics allows capturing key features that convective parameterizations struggle with. During the last 30 years, many studies focused on statistical response of cloud ensembles to the large-scale forcing over a limited area (e.g., Soong and Ogura 1980; Soong and Tao 1980; Tao and Soong 1986). So far, cloud-resolving models appear superior to any kind of convective parameterization, as found by comparing model results to observations. However, the computational cost still severely limits global cloud-resolving simulations and alternative approaches need to be explored. An important application of cloud-resolving modeling in the context of global simulation is validation and improvement of other approaches designed to estimate feedbacks from convective to mesoscale, synoptic, and global scales.

Rescaling approaches have also been suggested to extend cloud-resolving modeling to global simulations. The underlying idea is to artificially reduce the scale separation between convective and planetary scales, and thus to make explicit simulation of convection computationally feasible in global domains. The Diabatic Acceleration and Rescaling (DARE) approach (Kuang et al. 2005) and the hypo-hydrostatic approach (Pauluis et al. 2006; Garner et al. 2007) are examples of such techniques. In DARE, Earth’s diameter is reduced, the rotation rate is increased, and diabatic processes are accelerated. In the hypo-hydrostatic approach, the vertical acceleration is rescaled. Pauluis et al. (2006) have shown that both approaches are mathematically equivalent and they reduce the scale separation between convection and the planetary scale without affecting the dynamics at large scales. However, changes in the behavior of convection due to the rescaling limit the applicability of these methods. Nevertheless, they illustrate how mathematical rescaling can offer a computationally efficient way to use cloud-resolving models in global simulations.

The second approach is to take advantage of a cloud-resolving model for global simulations through the superparameterization methodology (Grabowski and Smolarkiewicz 1999; Grabowski 2001, 2004; Randall et al. 2003; Majda and Grooms 2014). In this framework, a two-dimensional cloud-resolving model with periodic lateral boundaries is embedded within each column of a global model to simulate interactions between convective and global scales. The simulated large-scale flow includes the convective feedback from small to large scales, and convective scales respond to the forcing from the large-scale dynamics. In the original superparameterization (SP), convective feedback is calculated using a cloud-resolving model applying horizontal domain equal (or approximately equal) to the large-scale model horizontal grid length. Grabowski (2001, section 3) simulated a 2D Walker cell of 4000-km horizontal extent applying the SP approach with different horizontal grid lengths (from 20 to 500 km) and thus different extents of the SP model horizontal domain. Results from these simulations were compared to the fully resolved simulations [described in Grabowski et al. (2000)] as well as between each other. SP seemed reasonably successful in reproducing large-scale conditions as simulated by the cloud-resolving model (e.g., dry subsidence and humid ascent regions, large-scale flow featuring the first and second baroclinic modes, etc.). However, mesoscale organization of convection and the strength of the quasi-two-day oscillations, the prominent feature of the fully resolved simulations, were significantly different between SP simulations.

Over the last 10 years, SP has been tested in many studies of tropical dynamics. Khairoutdinov et al. (2005) and DeMott et al. (2007) found that while Madden–Julian oscillation (MJO) is missing from the standard Community Atmosphere Model (CAM), it is simulated reasonably well with SP-CAM (i.e., the superparameterized CAM). They report several important improvements in simulating tropical climatology, such as a more realistic distribution of cirrus cloudiness or intense precipitation. However, some important biases persist, for instance, too heavy precipitation over the western tropical Pacific associated with the Indian monsoon or too low shallow-convection cloud fraction and light rain across parts of the tropics and subtropics. Studies attempting to explain the reason of excessive precipitation in the western Pacific during monsoon periods typically find significant correlation between moisture content in the column and precipitation. Thayer-Calder and Randall (2009) suggest that the difference comes from contrasting profiles of convective heating that excite different large-scale circulation (and thus affect surface wind and evaporative feedback) and subsequently differently moisten the troposphere. Luo and Stephens (2006) argue that convection–evaporation feedback is the main culprit of excessive rain and suggest that this may be due to the periodicity of SP’s cloud-resolving models leading to the prolonged presence of precipitating convection at a given location.

The mathematical aspects of the SP implementation are important as illustrated by the above examples and other studies (e.g., Grabowski 2004). Over the years, several algorithms have been proposed to implement the SP framework. Here, we evaluate the ability of the sparse space–time superparameterization (SSTSP) to accurately reproduce the interactions between convection and the large-scale flow. In the SSTSP framework, described in more detail in the next section, the embedded cloud-resolving model uses a horizontal domain that is small in comparison to the horizontal grid length of the large-scale model, and for a time period that is short when compared to the time step of the large-scale model. Here we apply SSTSP to perform systematic tests of impact of spatial compression applied sometimes in the previous SP implementations (Xing et al. 2009; Wang et al. 2011; Pritchard et al. 2011; Kooperman et al. 2013) and temporal acceleration. The latter one is similar to the DARE and hypo-hydrostatic rescaling, thus significantly increasing computational efficiency of the approach. As with the original SP, the goal of the SSTSP algorithm is to obtain a statistically correct representation of the convective impact on the large-scale flow at a reduced computational cost.

Preliminary SSTSP testing reported in Xing et al. (2009) applied two-dimensional simulations of an idealized squall line propagating in a periodic horizontal domain of 1024 km. The performance of the SSTSP algorithm was examined for a range of environmental conditions that differed in the prescribed vertical shear of the large-scale horizontal wind. The SSTSP algorithm seemed to capture propagation of the squall line and its speed. In particular, propagation speed appeared to be strongly controlled by the vertical profile of the large-scale shear, with no significant drawbacks of the SSTSP algorithm. Contrasting convective organizations were simulated for different shears, from the squall line to decaying convection. This provided hope for the SSTSP algorithm in simulations of different convective regimes for various large-scale conditions. Furthermore, structural agreement was found for large-scale features of simulated convective systems since pattern correlation was high for horizontal velocity or specific humidity. However, the impact of the SSTSP algorithm on large-scale features (e.g., the mean temperature and moisture profiles) was severely limited because of the short simulation time (36 h) and relatively small computational domain.

Here, we investigate the accuracy of the SSTSP algorithm in reproducing interactions between convection and the large-scale flow in an idealized Walker cell circulation. We compare SSTSP results against the benchmark solution obtained with the cloud-resolving model. The latter is described in more detail in Slawinska et al. (2014, hereafter SPMG) focusing on the intraseasonal variability of the Walker cell with the time scale of about 20 days. The low-frequency oscillation features four phases: the suppressed, intensification, active, and weakening. Intensification of the circulation is associated with the broadening of the large-scale ascent region, which in turn is strongly coupled to propagating synoptic-scale systems. Details of the SSTSP framework and its implementation are given in section 2. Results of simulations applying the SSTSP framework are discussed in section 3 and are compared to the results from SPMG’s cloud-resolving model. Section 4 provides a discussion of the model results and concludes the paper.

## 2. Model and experimental setup

In this study, we use the anelastic nonhydrostatic atmospheric model EULAG [Smolarkiewicz and Margolin 1997; see Prusa et al. (2008) for a comprehensive review]. EULAG applies finite-difference dynamics based on the multidimensional positive definite advection transport algorithm (MPDATA) scheme (Smolarkiewicz 2006), which is monotonic and intrinsically dissipative. The model is used without any additional subgrid-scale diffusion and a gravity wave absorber is added in the uppermost 8 km of the domain. The one-moment microphysical scheme of Grabowski (1998) that solves equations for cloud water and precipitation is applied here. These two classes correspond to either liquid (cloud water and rain) or solid (cloud ice and snow) condensate, for temperatures above 268 K or below 253 K, respectively, and mix of these two for the temperature range in between. Notably, condensation is calculated with no supersaturation allowed as defined with respect to either water or ice or a mix of the two, depending on the temperature. Also autoconversion, accretion, evaporation, and fallout of precipitation are taken into account.

Here, we apply the SP methodology (Grabowski 2001, 2004) and implement the SSTSP framework as described briefly below [see also Xing et al. (2009)].

### a. SP and SSTSP frameworks

#### 1) Large-scale and cloud-resolving model equations

*φ*variables:

*U*and

*u*) and vertical (

*W*and

*w*) velocities, potential temperature (Θ and

*θ*), water vapor (

*Q*

_{υ}and

*q*

_{υ}), condensed water/ice (

*Q*

_{c}and

*q*

_{c}), and precipitating water/ice (

*Q*

_{p}and

*q*

_{p}) mixing ratios, the latter two following the representation of moist thermodynamics of Grabowski (1998). Evolution of Φ and

*φ*can be symbolically written as

*S*

_{Φ}and

*S*

_{φ}represent various source terms in the large-scale and small-scale models, respectively (such as the buoyancy, pressure gradient, radiative cooling, surface fluxes, phase changes of the water substance, precipitation formation and fallout, gravity wave absorber, etc.). Surface fluxes and radiative transfer are calculated here in the large-scale model, while phase changes and precipitation formation/fallout are considered in the small-scale model only and they affect the large-scale fields through the small-scale feedback. In general, one needs to ensure that a given source is included only once between the two models, that is, no double counting takes place. The source terms

*S*

_{Φ}and

*S*

_{φ}need to be appropriately designed between the two models. For instance, the pressure gradient terms are independently formulated between the models (e.g., via the anelastic continuity equation). The variable

#### 2) Coupling procedure in SP

*X*× Δ

*Z*, contains a cloud-resolving (small scale) model that has

*N*

_{x}×

*N*

_{z}grid points of grid size Δ

*x*× Δ

*z*, for which

*T*, the evolution from time

*T*to

*T*+ Δ

*T*of the large-scale variable is calculated first:

*T*:

*T*+ Δ

*T*) and

*T*, as given by (10).

*T*+ Δ

*T*), the vertical profiles of large-scale forcing for the small-scale variables,

*N*

_{x}points of the small-scale model. With the large-scale forcing formulated as above and assumed constant for the large-scale time step, the small-scale model equations are advanced from

*T*to

*T*+ Δ

*T*:

*N*

_{t}time steps for which

#### 3) Sparse space–time algorithm

*p*

_{x}(i.e.,

*p*

_{x}smaller number of model columns; reduced space strategy) and the number of small-scale time steps by a factor of

*p*

_{t}(reduced time strategy). In such a case, the number of small-scale time steps in every large-scale time step and the number of columns in every large-scale grid,

*p*

_{t}(i.e., adding the time rescaling of the large-scale forcing), and applying horizontal averaging over the rescaled small-scale domain (marked

*φ*|

^{T+ΔT}, are assumed equal to the solution of (14) with accelerated forcing:

Elementary considerations [similar to those involving Eqs. (10) and (11) in Grabowski (2004)] document that the SSTSP algorithm outlined above ensures appropriate transfer of information between the small-scale and large-scale models despite spatial compression and temporal acceleration. For instance, if either *S*_{Φ} in (3) or *S*_{φ} in (4) is assumed constant, then the tendency due to this source is correctly passed from one model to another (i.e., from the large-scale to small-scale model for *S*_{Φ} and vice versa for *S*_{φ}) when spatial compression and temporal acceleration are applied.

Beyond mathematical consistency, one should be also aware of physical limitations of the SSTSP methodology. For the spatial compression, a small horizontal domain of the small-scale model may affect not only the statistical sampling of small-scale features, but their evolution as well, evolution of convective cells in particular. Since the mean vertical velocity within SP models at any level has to vanish (because of periodic lateral boundary conditions), the upward convective mass flux has to be balanced by the environmental subsidence. The key point is that the vertical development of convective clouds may be affected when the computational domain is reduced to a small number of columns. The temporal acceleration is perhaps more difficult to interpret. The approach taken in Xing et al. (2009) and followed here [cf. (16)] implies that the large-scale forcing is increased in proportion to the temporal acceleration factor *p*_{t}. The idea is that the original large-scale forcing has to be increased so the small-scale processes can appropriately respond over the *p*_{t}-shorter time. An alternative approach might be to keep the large-scale forcing unchanged, but instead increase the small-scale feedback by *p*_{t}. In other words, the small-scale response to the original feedback would be calculated only for the *p*_{t} fraction of Δ*T* and linearly extrapolated (i.e., increased by a factor of *p*_{t}) before being applied to the large-scale model. Such a procedure would lead to the same evolution in time of Ψ and *φ* for the case of a constant source. However, considering fundamental differences between time scales involved in small-scale and large-scale processes, extrapolation of the small-scale response seems more problematic than scaling up the large-scale forcing.

### b. Experimental design

The developments presented in the previous section are tested applying the Walker cell circulation in the two-dimensional domain following SPMG. As in SPMG, the environmental profiles come from a simulation of radiative-convective equilibrium applying a cloud-resolving model, the System for Atmospheric Modeling, with the National Center for Atmospheric Research (NCAR) CAM3 interactive radiation scheme (Khairoutdinov and Randall 2003). The planetary-scale circulation is driven by the surface fluxes and radiative cooling. The sea surface temperature (SST) distribution is given by a cosine-squared function, with 303.15 K in the center and 299.15 K at the periodic lateral boundaries. Radiative cooling is given by the average profile of radiative tendency in the radiative-convective simulation and by the relaxation term toward the equilibrium value of potential temperature with the 20-day time scale.

A cloud-resolving model has been set up with a 40 000-km horizontal scale and 24 km in the vertical. Uniform resolution of 2 km in the horizontal direction and 500 m in the vertical has been applied. The model has been run with a time step of 15 s for over 320 days, and the last 270 days have been analyzed. More detailed description of the modeling setup can be found in SPMG, which discusses results from the cloud-resolving simulation that provide the reference for SP simulations.

In SP simulations, the large-scale domain spans 40 000 km with horizontal and vertical grid lengths of 48 km and 500 m, respectively. The large-scale time step is 180 s. The cloud-resolving domain has horizontal and vertical grid lengths of 2 km and 500 m, respectively, and a small-scale time step of 15 s. The simulations are run for 340 days, and the last 290 days are analyzed. Because of the simulation length, no other SP setups [i.e., either larger or smaller large-scale model grid length; cf. section 3 of Grabowski (2001)] were considered. Simulations with various time accelerations and space compressions are compared. The horizontal domain of the cloud-resolving model is equal to the large-scale horizontal grid length (i.e., *p*_{x} = 1) or is reduced by a factor of 2 (*p*_{x} = 2) or 3 (*p*_{x} = 3). Also, for every large-scale time step, the time integration in the cloud-resolving domains is performed for the period either equal (*p*_{t} = 1) or 2 (*p*_{t} = 2), 3 (*p*_{t} = 3), and 4 (*p*_{t} = 4) times shorter than the large-scale model time step. A simulation with a given spatial compression (*p*_{x}) and temporal acceleration (*p*_{t}) will be referred to as “SSTSP*p*_{x}*p*_{t} simulation.” For instance, a simulation with *p*_{x} = 2 and *p*_{t} = 3 will be called “SSTSP23 simulation.” In total, 12 simulations are performed with different time accelerations and space compressions. We will refer to them as “SSTSP simulations.” SSTSP simulations are compared to the benchmark case obtained with the cloud-resolving model (CRM) and analyzed in SPMG, and referred to as the “CRM simulation” thereafter.

## 3. Results

SSTSP simulations reproduce the key characteristics of the CRM simulation. In particular, large-scale overturning circulation is simulated in the large-scale domain, with the large-scale ascent over warm pool and subsidence over cold SSTs. Similarly to the CRM simulation, variability across a wide range of scales is simulated. We start with a discussion of the mean state. Subsequently, we present analysis of high- and low-frequency variability, with the latter analyzed in more detail. The emphasis is on comparing the SP and CRM simulations (the latter one documented in detail in SPMG) and evaluating the impact of the spatial and temporal scaling factors *p*_{x} and *p*_{t}.

### a. The mean Walker cell circulation

Figure 1 shows the time-averaged horizontal velocity field for the CRM and SSTSP simulations (the former already shown in Fig. 2a in SPMG) as well as the difference between them. The CRM large-scale circulation features surface and midtropospheric mean horizontal flows toward the highest SST in the center of the domain. The horizontal velocity maxima are around 10 and 5 m s^{−1} at the surface and around 6-km altitude, respectively. The upper-tropospheric outflow from the center of the domain features maximum velocities of over 20 m s^{−1}. SSTSP simulations exhibit similar large-scale circulation, with low- and midlevel convergence accompanied by the upper-tropospheric divergence over warm SSTs, that is, with the first and second baroclinic modes. The most apparent difference between CRM and SSTSP simulations is the narrower ascending region in the center of the domain in SSTSP cases. Although the patterns and amplitudes of the horizontal flow are similar in all simulations, the difference plots between CRM and SSTSP show significant deviations that seem to increase with the spatial compression and temporal acceleration, with the SP simulation without compression and acceleration (i.e., SSTSP11) being the closest to CRM as one might expect. Although not shown in the figure, the differences depend primarily on the horizontal extent of the SP domains (i.e., they increase with the increase of *p*_{x}), and there seems to be no systematic impact of the temporal acceleration (i.e., increasing the *p*_{t} parameter).

Figures 2 and 3 document the impact of spatial compression and time acceleration on the mean (i.e., horizontal and time averaged) profiles of the potential temperature and water vapor mixing ratio. Figure 2 shows the difference between profiles from SSTSP with various spatial compressions and CRM. Mean profiles for the SSTSP11 simulation are close to CRM, and the differences increase with the spatial compression. The SSTSP31 simulation features up to 8-K colder upper troposphere and up to 2 g kg^{−1} lower moisture in the lower troposphere when compared to CRM. The relative humidity profiles (not shown) agree relatively well below 8 km for all simulations and differ significantly above 10 km, with no obvious sensitivity to the spatial compression. The differences between the temperature profiles are consistent with a heuristic argument that reducing the horizontal extent of SP computational domains (i.e., increasing *p*_{x}) makes convective overturning more difficult and leads to a colder upper troposphere. The water vapor difference profiles can be explained by a narrower ascending region in the center of the domain as illustrated in Fig. 1 and further quantified below. As shown in Fig. 3, time acceleration leads to the mean temperature (moisture) profiles that are warmer (more humid), but the effects are significantly smaller than for the spatial compression, especially for the moisture.

Figures 4 and 5 show spatial distributions of the difference between the SSTSP and CRM simulations for the mean temperature and water vapor mixing ratio, respectively. The differences are averaged over days 50–340. For the temperature, the patterns are dominated by the differences in the mean profiles (cf. Fig. 2), with small gradients between regions with high and low SST (i.e., mean ascent and mean subsidence). In the CRM simulation, the temperature field at a given level is homogenized by convectively generated gravity waves that maintain a small horizontal temperature gradient. Such a mechanism is also efficient in SP simulations, including SSTSP, as documented by the relatively small horizontal temperature gradients in Fig. 4. The water vapor field, on the other hand, can only be homogenized by the physical advection and the differences between SSTSP and CRM simulations are larger, as shown in Fig. 5. The largest differences (in the absolute sense) are near the center of the domain, likely because of the different width of the central ascending region and differences in the large-scale circulation (cf. Fig. 1). The differences increase with the increase of the spatial compression and temporal acceleration. The lower troposphere above 1 km is drier in SSTSP than in the CRM, in both the ascent and subsidence regions, perhaps with the exception of the narrow zone over the coldest SSTs. The level of maximum difference outside the central region at heights between 2 and 3 km corresponds to the low-level cloud tops (see below). The upper troposphere is drier at the warm pool edges, likely because of the narrower region of deep convection in the SSTSP simulations.

Figures 6 and 7 show time-averaged mean fields and profiles of the cloud condensate mixing ratio, respectively. Figure 6 shows that shallow convection occurs over the entire domain, while deep convection is confined to the warm pool. The region with deep convection narrows when the spatial compression and temporal acceleration increase. There are also systematic changes of the mean cloud condensate profiles as documented in Fig. 7. The figure documents the classical trimodal characteristics of the tropical moist convection: shallow, congestus, and deep (cf. Johnson et al. 1999), with the lower-tropospheric maximum associated with shallow convective clouds, and mid- and upper-tropospheric maxima marking detrainment levels from congestus and deep convection, respectively. Temporal acceleration results in a significant shift of the profiles toward higher values (a factor of approximately 2 between Figs. 7a and 7d). Spatial compression for a given temporal acceleration has a relatively smaller effect, with a systematic decrease of cloud condensate above 5 km.

Figure 8 shows mass flux profiles for CRM and SSTSP simulations with various spatial compressions and temporal accelerations. Since these profiles are derived by averaging the cloud-model data, they represent the impact of the SSTSP methodology on convective transport. The SP simulation with neither spatial compression nor temporal acceleration (i.e., SSTSP11) gives the mean mass flux close to the one from the CRM simulation. Spatial compression (i.e., SSTSP31) leads to significantly modified vertical velocity distribution (see Fig. 9) and thus reduced mass flux, arguably because of the impact of a reduced extent of the cloud-model computational domain on the convective transport as argued at the end of section 2a. In contrast, temporal acceleration (i.e., SSTSP13) leads to a significant increase of the mass flux, arguably because of the increase of the large-scale forcing [cf. (7) and (13)]. Combining spatial compression and temporal acceleration (i.e., SSTSP33) results in the convective mass flux in between the ones obtained from the simulations with either spatial compression or temporal acceleration.

The differences in the convective mass flux affect the mean (domain and time averaged) profiles of the precipitation water mixing ratio as shown in Fig. 10. The simple microphysics parameterization used in the simulations assumes precipitation to be in the form of snow (rain) in the upper (lower) troposphere, with snow sedimenting with significantly smaller vertical velocity. This explains the difference between lower- and upper-tropospheric values of each profile. However, the magnitude of the profiles [i.e., the largest (smallest) for SSTSP13 (SSTSP31)] is in direct response of the convective mass flux shown in Fig. 8. The difference between precipitation water profiles might have a significant impact on model results once an interactive radiation scheme is used in place of a prescribed radiative cooling applied in current simulations.

The SSTSP framework significantly modifies the spatial distribution of convection and related statistics. The differences in cloudiness are associated with different spatial distributions of the time-averaged precipitable water content, cloud-top temperature, and precipitation rate, as shown in Fig. 11, respectively, with their mean values given in Table 1. As the figures document, SSTSP simulations are characterized by significantly narrower distributions of all the quantities. In the CRM simulation, the central 10 000 km is characterized by the mean cloud top temperature around 288 K, precipitable water around 75 kg m^{−2}, and surface precipitation around 0.45 mm h^{−1}. All distributions are relatively flat and feature steep gradients at the edges of the warm pool with the mean precipitation dropping below 0.1 mm h^{−1} and mean cloud-top temperature increasing to around 300 K. SSSTP simulations, on the other hand, are characterized by narrow distributions, with peaks at the center and steep gradients of the mean cloud-top temperature and precipitation. These differences also occur in SSTSP11, that is, the SP simulation with neither time acceleration nor spatial compression, and thus are a general feature of the SP simulation.

Mean precipitable water content (Prw; kg m^{−2}), cloud-top temperature (Cltop; K), and precipitation (Precip; mm h^{−1}).

Because of the complicated impact of the time acceleration on diabatic processes, an intrinsic feature of the SSTSP framework, it is impossible to rescale the cloud-top temperature between CRM and SSTSP simulations. Increasing temporal acceleration leads to more intense convective activity (cf. Fig. 8), increased cloudiness and precipitable water, and a decreased mean cloud-top temperature (cf. Table 1). These aspects of temporal acceleration have been pointed out by Pauluis et al. (2006) and they seem to be related to the way microphysical processes (in particular fallout of rain) are handled. No acceleration of microphysical processes is applied here, potentially impacting the balance between processes responsible for moistening and drying the troposphere.

In summary, the SSTSP framework appears to simulate the 2D Walker circulation qualitatively well. In particular, the mean large-scale flow consists of a deep overturning circulation (first baroclinic mode) and a midtropospheric jet (second baroclinic mode). Deep convection occurs primarily over the warm pool, and subsidence regions are dominated by shallow convection. However, detailed comparison reveals systematic differences in the model mean state. These differences are mainly artifacts of the original implementation of SP with additional drawbacks introduced along with SSTSP framework. The artificial scale separation between large-scale and small-scale models and the periodicity of small-scale models impose significant limitations on the flow field in the small-scale domain (e.g., vanishing mean mass flux) and subsequently on the simulated convection and its organization. Temporal acceleration of large-scale forcing is associated, in turn, with enhancement of convection of which nonlinear nature appears to be an obstacle for proper scaling of cloudy properties, impacting significantly, for example, the radiative calculation. Moreover, the convective feedback to the large scale cannot be easily modified, which is associated with a negative impact on mean large-scale conditions.

### b. Transients

SPMG document several transient features occurring in CRM simulation. The large-scale flow is characterized by low-frequency variability featuring 20-day oscillations with alternating periods of strong and weak overturning circulation. The strong circulation phase is associated with intense convection and expansion of the large-scale convergence region over the warmest SSTs. The weak circulation phase, on the other hand, features reduced convective activity and a narrower convergence region. The expansion (compression) of the convergence region coincides with synoptic-scale convective activity propagating from (to) the center of the domain with the average speed between 5 and 10 m s^{−1}.

Here, we investigate if the SSTSP framework is capable of capturing these oscillations. Figure 12 shows Hovmoeller diagrams of cloud-top temperature for SSTSP11 and SSTSP33 simulations. The figure shows that the variability in SP simulations is of a similar character to that in the CRM simulation. In particular, the zigzag pattern formed by very cold tops of convective cloud systems propagating toward and then away from the convergence region apparent in the CRM simulation can also be noticed in the SP model. However, the coherency decreases with the increase of spatial compression and temporal acceleration. This is consistent with the fact that coherent propagation of convective-scale features across the SP model grid is more difficult than in the CRM model because the cloud-scale models only communicate through the large-scale model dynamics. Another feature apparent in Fig. 12 is that convection seems to be more localized in the center of the domain as already documented in Figs. 11 and 12.

An inability to simulate an organized convective system in climate models has been documented before (Pritchard and Somerville 2009a,b). Just recently propagation of convective systems has been observed in SP-CAM3.5 and SP-CAM5 by Pritchard el al. (2011) and Kooperman et al. (2013). In their study, they reproduce nocturnal convection crossing the Great Plains east of the Rocky Mountains in a specific implementation of SP-CAM. In particular, they find it sensitive to such factors as microphysical parameterization, horizontal resolution of a cloud-resolving model, and the choice of the dynamical core, with no detailed explanation of these dependencies. Although they attribute the propagation of mesoscale convective systems to a large-scale first-baroclinic wave driven by convective heating, they do not exclude other mechanisms (e.g., large-scale advection of water vapor). As such, an understanding of propagation of convective systems remains elusive and more studies are needed to overcome that. Here we apply SSTSP and systematically modify the coupling of the large-scale and the cloud-resolving model, finding the propagation of convective systems impacted negatively both by spatial compression and temporal acceleration. We hope to report in the future on the details of physical mechanisms associated with these dependencies.

### c. Low-frequency variability

Low-frequency variability in the CRM simulation has been analyzed in detail in SPMG. There, we apply the empirical orthogonal function (EOF; von Storch and Zwiers 1999) analysis and develop an index of the low-frequency variability. Subsequently, we construct a composite of low-frequency variability with a lag-regression analysis applied to the index. We analyze reconstructed fields of different dynamical variables and describe the low-frequency oscillation. We find the low-frequency variability of a 20-day period, triggered by anomalously intense deep convection over the warm pool. This, in turn, is the consequence of large-scale horizontal advection of anomalously moist air from the subsidence region after the period of moisture buildup through anomalously intense shallow convection.

Here, we investigate if the low-frequency variability is captured by applying the same methodology as for the CRM simulation in SPMG. First, we perform EOF analysis for the last 290 days of large-scale surface wind data with 1-h temporal resolution. Subsequently, for every SSTSP simulation, we analyze the low-frequency variability by applying the principal component of the leading EOF (see section 4 in SPMG). Table 2 presents the main characteristics of the leading EOF for various SSTSP simulations and for the CRM simulations from SPMG. All SSTSP simulations exhibit a dominant mode of low-frequency variability corresponding to a strengthening/weakening of the low-level flow as identified previously for the CRM simulation in SPMG. The power spectrum peaks for the period in between 23 and 26 days and compares reasonably well with the 20-day period of the CRM simulation. It thus appears that the SSTSP framework captures the variability corresponding to the intraseasonal band that, in turn, is responsible for a significant percent of the total variance as in the CRM simulation. Overall, SSTSP simulations with a larger spatial compression or temporal acceleration tend to exhibit lower total variance. SSTSP23 and SSTSP24 simulations feature the closest variance to the CRM simulation.

Eigenvalue periods of the first EOF for the surface wind time series and standard deviation of its principal component. Simulations with SSTSP algorithm (with *p*_{x} and *p*_{t} as given in the first column). CRM results are included in the bottom row.

*f*(

*x*,

*t*) against the first EOF as follows:

*σ*stands for the standard deviation of the first EOF, PC

_{1}. Finally, we obtain the composite by adding the mean value of the given field

All SSTSP simulations reproduce phases of the low-frequency oscillation, with the exemplary composite of the horizontal velocity for SSTSP22 simulation shown in Figs. 13 and 14. Figure 13 can be compared to Fig. 8 in SPMG, whereas Fig. 14 can be compared to Figs. 9a to 12a in SPMG. As in SPMG, the low-frequency oscillation consists of four phases: suppressed, strengthening, active, and decaying. The mechanisms behind low-frequency oscillation are robust and are reproduced in all SSTSP simulations. As in the CRM simulation, large-scale advection of moisture is correlated with oscillations of convective activity and large-scale circulation. The suppressed phase is characterized by weak large-scale overturning circulation and decreasing deep convective activity in the central part of the domain. This, in turn, is associated with a drier troposphere due to an anomalously strong midtropospheric advection of dry air from the subsidence region and an anomalously weak advection of the moist surface air to the central part of the domain. At the same time, anomalously weak subsidence allows for moisture buildup over the subsidence region, as shallow convective activity intensifies. Circulation strengthens as low-tropospheric anomalously moist air is advected into the central part of the domain and deep convection intensifies. As deep convective activity reaches its peak, the troposphere warms and dries as a result of the latent heat release and intense precipitation. The decaying phase follows when the central region dries out because of the intense precipitation, and it is accompanied by a strong midtropospheric jet bringing dry air from the subsidence region and advection of anomalously dry low-tropospheric air as the shallow convection weakens due to strong subsidence.

## 4. Discussion and conclusions

The primary purpose of this paper is to evaluate the sparse space–time superparameterization (SSTSP) introduced in Xing et al. (2009). SSTSP extends the original superparameterization (SP) approach, where convective processes are simulated explicitly by a cloud-resolving model embedded in every large-scale model column. The motivation behind SSTSP comes from the quest for computationally affordable and statistically accurate simulations of large-scale circulation that crucially depend on convective activity. SSTSP addresses this issue by significantly reducing cloud-resolving calculations and at the same time assuring its statistically accurate small-scale (convective) feedback. The feedback is obtained by rescaling statistics from cloud-resolving calculations over time and horizontal domain spans that are reduced relative to the large-scale time and space resolutions.

Xing et al. (2009) performed the initial tests of the SSTSP methodology. They conducted idealized simulations of a squall line within 1000-km horizontal domain for a 6-h period. It was shown that SSTSP captures propagation of a squall line across the domain, with the propagation speed controlled by the prescribed shear. Here, we evaluate the SSTSP framework for an idealized Walker cell setup. This is a more complex case than in Xing et al. (2009) because it features a wider range of spatiotemporal scales, up to planetary scales and time periods up to several tens of days. In contrast to Xing et al. (2009), larger-scale flow can evolve in response to feedbacks from moist convection, and in turn it can affect the subsequent convective development. We evaluate the performance of the SSTSP algorithm by comparing solutions to those obtained applying the cloud-resolving model (CRM) described in SPMG.

We find that SSTSP is capable of reproducing key characteristics of the cloud-resolving Walker cell simulation. In agreement with CRM results, the mean state features the first and second baroclinic modes with deep convection over high SST organized into propagating systems. The properties of these convective systems (e.g., structure, propagation) are affected by the horizontal extent of SP cloud-resolving domains. This is an artifact of the periodicity of embedded cloud-resolving models that cannot be avoided. SSTSP captures intraseasonal variability predicted by the CRM model that consists of four distinctive stages—suppressed, intensification, active, and weakening—along with the mechanisms driving them. Differences in convective evolution and propagation result in some differences between SSTSP and CRM simulations, such as in the mean sounding, spatial distribution of the cloudiness, and surface precipitation, or in the spatiotemporal characteristics of the low-frequency oscillation. Differences in the mean cloudiness (cf. Fig. 9) will most likely be accentuated when an interactive radiation transfer scheme is used, an aspect not addressed in the current study.

Numerical simulations discussed here can be put into the context of those discussed in Grabowski (2001, hereafter G01). Section 3 of G01 presented SP simulations of a 2D flow driven by large-scale SST gradients, although of a significantly smaller horizontal extent (a computational domain of 4000 km in G01 rather than 40 000 km here and in SPMG). The SP simulations in section 3 of G01 were compared to CRM simulations discussed in Grabowski et al. (2000). G01’s SP simulations used various horizontal grid lengths of the large-scale model (from 20 to 500 km; referred to as P20 and P500, respectively) with the horizontal extent of the embedded CRM model matching the large-scale model grid length. G01’s results documented a significant impact of the specific setup of the SP model configuration (i.e., from P20 to P500), especially for the mesoscale convective organization (cf. Fig. 8 therein). In contrast, the SP simulations presented here feature just a single 48-km horizontal grid length of the large-scale model and explore the impact of SSTSP methodology. Such a grid length can be argued to follow a recommendation of Grabowski (2006a) who suggested that the SP approach is better suited for large-scale models with horizontal grid lengths in the mesoscale range (i.e., a few tens of kilometers). This is because, in the mesoscale grid length case, the embedded SP models represent the effects of small-scale convective motions only, and the convective mesoscale organization (e.g., into squall lines) can be simulated by the large-scale model. Both convective and mesoscale circulations have to be represented by the SP model when the large-scale model grid length is hundreds of kilometers, as in typical global climate applications (e.g., Khairoutdinov et al. 2005; DeMott et al. 2007, among others). Section 4 of G01 applies the SP methodology to the problem of large-scale convective organization on an idealized constant-SST (“tropics everywhere”) aquaplanet. Although not emphasized there, the SP aquaplanet simulations [as well as subsequent studies, e.g., Grabowski (2006b)] already apply the spatial compression methodology because of the disparity between the large-scale model grid length and the horizontal domain size of the embedded CRM model.

The obvious main drawback of the SP approach is that every cloud-resolving model is independent of each other and communicates solely by the large-scale model dynamics (i.e., through the large-scale forcings). The key point is that cloud systems cannot propagate directly from one large-scale grid box to the other, but they remain locked in a single large-scale grid box because of the periodic lateral boundary conditions. As discussed by Jung and Arakawa (2005), periodic lateral boundary conditions require the mean mass flux to vanish. As a result, updrafts get weaker as the horizontal extent of the cloud-resolving domain decreases. Large-scale thermodynamical fields are also modified, for instance, the lower (upper) troposphere becomes moister (drier). This key drawback of the SP (and thus SSTSP) approach was also noted in other studies (e.g., G01) and it is evident in our simulations. To overcome these limitations, Jung and Arakawa (2005) suggest an alternative approach where periodic boundary conditions are abandoned and the adjacent cloud-resolving models are linked allowing for the direct propagation of small-scale perturbations from one large-scale model grid box to another. Such an approach is relatively straightforward in a 2D large-scale model framework, but it requires more complicated parallel-processing methods as opposed to the “embarrassingly parallel” logic of the original SP. Perhaps more importantly, this simple idea leads to a significantly more complex methodology when implemented into a 3D large-scale model (cf. Jung and Arakawa 2010, 2014). Considering these factors, we feel that the traditional SP/SSTSP methodology can still serve as a valuable technique in large-scale models featuring mesoscale horizontal grid lengths and variable orientation of SP cloud-resolving models (cf. Grabowski 2004). We hope to report on such numerical experiments in forthcoming publications.

## Acknowledgments

J. Slawinska acknowledges funding from CMG Grant DMS-1025468 as well as support from the Center for Prototype Climate Modeling at NYU Abu Dhabi. This research was carried out on the high performance computing resources at New York University Abu Dhabi.

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