Convergence of Convective Updraft Ensembles With Respect to the Grid Spacing of Atmospheric Models

Atmospheric deep moist convection can organize into cloud systems, which impact the Earth's climate significantly. High‐resolution simulations that correctly reproduce organized cloud systems are necessary to understand the role of deep convection in the Earth's climate system. However, there remain issues regarding convergence with respect to grid spacing. To investigate the resolution necessary for a reasonable simulation of deep convection, we conducted grid‐refinement experiments using state‐of‐the‐art atmospheric models. We found that the structure of an updraft ensemble in an organized cloud system converges at progressively smaller scales as the grid spacing is reduced. The gap between two adjacent updrafts converges to a particular distance when the grid spacing becomes as small as 1/20–1/40 of the updraft radius. We also found that the converged inter‐updraft distance value is not significantly different between Reynolds‐averaged Navier–Stokes simulations and large eddy simulations for grid spacings in the terra incognita range.


Introduction
Deep convective clouds in the atmosphere have a major impact on climate sensitivity (Bony et al., 2015). However, there are still issues with their representation in climate models, causing great uncertainty in climate prediction (Mauritsen & Stevens, 2015;Stevens & Bony, 2013). Convective clouds often form selforganized systems with spatially hierarchical structures, such as tropical cyclones or super cloud clusters associated with Madden-Julian oscillations (MJO; Madden & Julian, 1972). This self-organization of convective clouds makes it difficult to assess their impact on the climate (Bony et al., 2015;Mauritsen & Stevens, 2015). For a better understanding of the behavior of cloud systems and their roles in the global circulation, numerical models that can simulate cloud systems from the elementary processes of atmospheric convection (i.e., cloud microphysics and turbulence) are useful tools. However, the convergence with respect to grid spacing remains debated. Here, convergence is defined as the state in which the structure of the simulated deep convection does not substantially change with further grid-spacing reductions.
Previous studies have shown that at least a 4-km horizontal grid spacing is necessary to simulate the structure of deep convective systems plausibly (Weisman et al., 1997), a 250-m grid spacing is required to resolve ©2019. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. isotropic turbulent flow in deep convection (Bryan et al., 2003), and a 200-m grid spacing is needed to achieve convergence for the vertical velocity profiles in deep convective updrafts (Khairoutdinov et al., 2009). Large scale heating and moistening due to deep moist convection are known to converge at around a 1-km grid spacing both for idealized (Panosetti et al., 2018) and real cases with complex terrains (Langhans et al., 2012;Panosetti et al., 2019). The convergence of large-scale tendencies is referred to as "bulk convergence" (Langhans et al., 2012), the achievement of which implies climate simulation capability of convection-permitting models with kilometer-scale grid spacing. On the other hand, "structural convergence," defined as the convergence of the statistics of individual cloud-scale structures (Panosetti et al., 2018), is known to not have been achieved even at a subkilometer grid spacing. Miyamoto et al. (2013Miyamoto et al. ( , 2015 investigated the grid-spacing dependence of the distance between the updraft cores of deep moist convection in a global model; however, convergence was not obtained even at an 870-m grid spacing. Panosetti et al. (2018Panosetti et al. ( , 2019 showed that a 550-m grid spacing is insufficient for the structural convergence of deep convection, unlike the bulk convergence. Parameter tuning of coarse-resolution models whose convergence is achieved only in the bulk sense does not guarantee the predictability of unknown future climate change. Comparing a converged structure of organized cloud systems with its natural counterpart is fundamental for improving their representation in our state-of-the-art atmospheric models. In this sense, the structural convergence of deep convection is important not only for short-term weather prediction, focusing on cloud-scale phenomena such as localized heavy rainfall and severe winds, but also for constructing reliable climate models. We explored the structural convergence of deep convection by systematically performing grid refinement experiments. We focused on convective updraft as in our previous studies (Miyamoto et al., 2013. Jeevanjee (2017) showed that the average vertical velocity of convective parcels converges at a 250-m grid spacing, while the average parcels' aspect ratio does not converge even at sub-100-m grid spacing; that is, the convergence of deep convective updraft is achieved in the bulk sense but not in the structural sense. This paper shows that the simulated convective system contains updraft ensembles with multiple-scale structures and converges at progressively smaller scales as grid spacing is reduced.

Experimental design
The simulation target was deep convection associated with the MJO in late November 2011. This was a typical MJO event observed during the CINDY2011/DYNAMO international field campaign (Yoneyama et al., 2013). See Figure 1 for an overview of experimental design and the sequence of procedures. First, we conducted global simulations with a horizontal grid spacing of 3.5 km using the Nonhydrostatic ICosahedral Atmospheric Model (NICAM; Tomita & Satoh, 2004; see also Satoh et al., 2014), which has been shown to yield realistic MJO simulations (Miura et al., 2007;Miyakawa et al., 2014). The initial conditions were constructed from the final operational global analysis data at 00:00 UTC on 21 November 2011, provided by the National Centers for Environmental Prediction. Next, an active convective region was downscaled using the Scalable Computing for Advanced Library and Environment-Regional Model (SCALE-RM; Nishizawa et al., 2015;Sato et al., 2015;Yoshida et al., 2017). In the downscaling experiments, we performed two series of experiments using one-way online nesting systems: a coarse-resolution experiment with grid spacings from 3,200 to 200 m (Exp-A), and a fine-resolution experiment with grid spacings from 800 to 50 m (Exp-B). The initial and boundary conditions of Exp-A were constructed from the NICAM output and those of Exp-B from the output of Exp-A with 800-m grid spacing.

Model Physics
In the NICAM simulation, microphysical processes, boundary layer turbulence effects, and radiation effects were parameterized using a six-class single-moment scheme (Tomita, 2008), Level 2 closure of the improved Mellor-Yamada scheme (Mellor-Yamada-Nakanishi-Niino [MYNN] scheme; Nakanishi & Niino, 2004, 2006, and mstrnX scheme (Sekiguchi & Nakajima, 2008), respectively. In Exp-A and Exp-B by SCALE-RM, the same microphysics and radiation schemes as in the NICAM simulation were used. Except for the grid spacing, domain size, and simulated time, the main difference between Exp-A and Exp-B was in the representation of the subgrid scale (SGS) turbulence. In Exp-A, a Level 2.5 closure of the MYNN scheme, which is based on a prognostic equation for turbulent kinetic energy, was used, whereas the Smagorinsky-type scheme (Brown et al., 1994;Lilly, 1962;Scotti et al., 1993;Smagorinsky, 1963) was used in Exp-B. The MYNN scheme is based on the Reynolds-averaged Navier-Stokes equations which is usually applied to horizontal grid spacings of more than 1 km and only calculates the SGS turbulent mixing in the vertical direction. That is, the horizontal diffusion is not explicitly treated in Exp-A. On the other hand, the Smagorinsky scheme is used for a large eddy simulation with sub-100-m grid spacing and calculates SGS turbulent mixing in three dimensions. SGS turbulence parameterization for intermediate grid spacing has been referred to as a "terra incognita" (Wyngaard, 2004) and remains a controversial issue in the modeling community (e.g., Honnert et al., 2016;Ito et al., 2015;Kitamura, 2016;Shin & Hong, 2015). Our experiments applied both the MYNN and Smagorinsky schemes to the grid spacing in  Tables S1 and S2 in the supporting information.

Algorithms for Detecting Deep Convective Updraft Cores
Our study used two different algorithms to detect separate deep convective updrafts. The first algorithm (Algorithm 1) is similar to the procedure used in our previous studies (Miyamoto et al., 2013. First, deep cloud regions are determined using the ISCCP simulator (Klein & Jakob, 1999;Webb et al., 2001), a satellite simulator for estimating the optical thickness, τ, and cloud top pressure, p top , of model columns. In this study, deep cloud regions were defined as regions where τ ≥ 60 and p top < 440 hPa. Next, the vertically averaged vertical velocity from the surface to the cloud top (w cloud ) was calculated in the deep cloud regions, where the cloud top was defined as the uppermost model level whose optical thickness from the model top is larger than 0.1. Finally, each updraft core is defined as a local w cloud maximum with the provision that w cloud is equal to or larger than a particular threshold value, w 0 . In the second algorithm (Algorithm 2), each updraft core is defined as a continuous region where w cloud is equal to or larger than the same w 0 used in Algorithm 1. The central position of each updraft core was defined as the "center of gravity" of w cloud and used to calculate the distance between updraft cores. Schematic illustrations of the two algorithms are shown in Figure S1. The threshold value, w 0 , was set to 0.5 m/s so that the mean vertical velocity profiles at the outer edges of updraft cores reach at least 1 m/s in the middle of the troposphere (see Text S1 and Figure S2 in the supporting information). Figure 2 shows an example of updraft cores extracted from a simulation snapshot. Although the updraft cores detected by Algorithm 1 are just points (Figure 2a), those detected by Algorithm 2 are two-dimensional objects; therefore, each core has a different area (Figure 2b). These two types of updraft core are qualitatively different. Therefore, the convergence behavior is also expected to be different. Figure 3 shows the grid-spacing dependence of the updraft cores detected by Algorithm 1, including the number of updraft cores per 10,000 km 2 deep cloud region (Figure 3a) and the mean value of the

Geophysical Research Letters
logarithm of the distance, d, between two nearest-neighbor cores, denoted by mean Log(d) (Figure 3b). The logarithm of d was used because its distribution has a strong positive skewness ( Figure S3). We found that the number density of the cores continued to increase up to a 50-m grid spacing (Figure 3a), that is, convergence for the updraft cores was not achieved when we reduced the grid spacing further than in our previous studies (Miyamoto et al., 2013. The mean Log(d) also showed no convergence (Figure 3b). The corresponding distance for mean Log(d) gradually increases from 5 to 8.5 times the grid spacing as the resolution increases. The intervals of adjacent updraft cores detected by Algorithm 1 are close to the scale of the so-called "effective resolution" (7 times the grid spacing) where the kinetic energy spectrum falls off the −5/3 slope in numerical atmospheric simulations (Skamarock, 2004). Algorithm 1 probably detects all the small structures with a size below the physically meaningful updraft structure. Therefore, the distance will not converge as long as we use this algorithm. Interestingly, the mean Log(d) for Exp-A and Exp-B are almost indistinguishable for grid spacings between 800 and 200 m, although Exp-A is less diffusive because of the absence of horizontal SGS turbulent mixing (see section 2.2).

Results Obtained Using Algorithm 2
Unlike Algorithm 1, Algorithm 2 provides information about the two-dimensional size of each detected updraft core. Figure 4 shows the data obtained using Algorithm 2, similarly to Figure 3. In Figure 4, the statistics for different size categories are shown. Each updraft core is categorized by its effective radius, R, which is defined as the square root of area S divided by π: R ≡ (S/π) 1/2 . We removed the updraft cores with R ≥ 8 km from the analysis because their sample size is too small for statistical analysis. Figure 4a shows that the number density for each size category reaches its maximum at different grid spacings, except for R < 0.5 km. We found that the maximum point scales perfectly with the effective radius of each size category: those for R = 4-8, 2-4, 1-2, and 0.5-1 km are located at grid spacings of 1,600, 800, 400, and 200 m, respectively. Figure 4b shows that the variation in mean Log(d) with respect to the grid spacing also scales with the effective radius. The minima of mean Log(d) for R = 4-8, 2-4, 1-2, and 0.5-1 km are located at grid spacings of 800, 400, 200, and 100 m, respectively. At the minimum, the horizontal scale of the updraft core corresponds to the effective resolution of the model: R is 5-10 times as large as the grid spacing. The mean Log(d) increases as the grid spacing is further reduced from the minimum point, indicating that the minimum is not the convergence point. For R = 4-8 and 2-4 km, convergence seems to occur at 200-and 100-m grid spacings, respectively. Therefore, a grid spacing as small as 1/20-1/40 of R is required for convergence. This  Figure S4). Assuming a simple scaling law, we can speculate that mean Log(d) for R = 1-2 km and 0.5-1 km will converge at grid spacings of 50 m and 25 m, respectively. As with the number density, the smallest category (R < 0.5 km) does not reach the minimum. However, the minimum and convergence points can be estimated from the above scaling law if it is further classified into R = 250-500 m, 125-250 m, and so on. We conclude that the structure of updraft ensemble in deep moist convection converges at progressively smaller scales in accordance with the scaling law with respect to the horizontal scale of the convective updraft.

Discussion
For the results of both algorithms, the mean Log(d) values for Exp-A and Exp-B are not significantly different in the terra incognita range. In the case of the updraft core of R = 4-8 km detected by Algorithm 2, the confidence intervals of 99% for Exp-A and Exp-B overlap from the minimum point (800-m grid spacing) to the convergence point (200-m grid spacing; Figure 4b). In the simulations, horizontal diffusion by the SGS turbulence model acted only in Exp-B. By contrast, the implicit numerical diffusion associated with the advection scheme acted in both Exp-A and Exp-B because we used third-order upwind spatial discretization (Table S2). If the implicit numerical diffusivity rather than the SGS turbulent diffusivity was dominant in the terra incognita range, a comparison of Exp-A and Exp-B would be meaningless. To assess this concern, we compared the impacts of implicit numerical diffusion and SGS turbulent diffusion using a simple scale analysis. According to Nishizawa et al. (2015), the time scale of implicit numerical diffusion, T num , for the thirdorder upwind spatial discretization can be estimated as T num~1 2L 4 /UΔx 3 , where L is the horizontal scale of variation, U is the wind speed, and Δx is the grid spacing. Meanwhile, the time scale of SGS turbulent diffusion, T SGS , can be estimated as T SGS~L 2 /ν SGS , where ν SGS is the coefficient of eddy viscosity. Thus, the ratio of T SGS to T num becomes (1) Figure 5 shows the change in T SGS /T num with respect to the horizontal scale of variation L. We assumed U = 10 m s −1 and used the median ν SGS of the whole analysis domain and period to estimate Equation (1). Note that the SGS turbulence model used in this study acts only when the pointwise Richardson number, Ri, is less than 0.25, and thus, we considered only the grid satisfying Ri < 0.25 for the determination of the median ν SGS . If the horizontal scale of variation is larger than the effective resolution (L ≈ 10Δx), T SGS /T num ≪ 1 ( Figure 5). In this case, the time scale of diffusion by SGS turbulence is much shorter than that of the numerical viscosity, and thus, the impact of SGS turbulent diffusion is much larger than that of implicit numerical diffusion. For the convergence point of the updraft core with R = 4-8 km, where R is 20-40 times as large as the grid spacing (Figure 4b), we can conclude that the converged updraft structures are not significantly different between models with the MYNN and Smagorinsky schemes despite their different horizontal diffusivities. Therefore, the MYNN scheme can be used beyond the theoretically applicable range, provided that the focus is on the spatial arrangement of convective updrafts having a horizontal scale of around 4 km or larger. However, differences in the turbulence parameterization probably change the vertical moisture flux within the planetary boundary layer and, consequently, change the total area of the convective region. This issue will be addressed in our next study.

Conclusions and Remarks
We conducted grid-refinement experiments on the simulation of deep moist convection associated with an MJO event to investigate the structural convergence of a convective updraft ensemble in an organized cloud system. We clarified that the structure of an organized cloud system reaches convergence at progressively smaller scales. A grid spacing as small as 1/20-1/40 of the updraft effective radius is required for the statistical properties of convective updrafts converging. We conducted both Reynolds-averaged Navier-Stokes simulations and large eddy simulations using the MYNN and Smagorinsky schemes, respectively, finding a not significantly different convergence behavior. Thus, the MYNN scheme can be applied to grid spacing in the terra incognita range if we focus on the spatial arrangement of convective updrafts having a horizontal scale of several kilometers.
Our results yield guidelines for the required horizontal grid spacing by comparing simulated convection structures with a particular spatial scale in the current model to their counterparts in nature. The same procedure can be applied to other model frameworks, for example, different microphysics and turbulence parameterizations, and other types of organized cloud systems, such as tropical cyclones. The scaling law with respect to the effective radius offers hope for the construction of new statistical models for the hierarchical structure of organized cloud systems. The nature of organized cloud systems could be captured by satellite, for example, Tropical Rainfall Measuring Mission (Huffman et al., 2007) and Global Precipitation Measurement (Hou et al., 2014), ground-based weather radar, and aircraft observation (e.g., LeMone & Zipser, 1980). Comparing these observations with high-resolution simulations could lead to improving numerical atmospheric models, which are required for understanding the essential role of deep moist convection.

Acknowledgments
We thank the three anonymous reviewers of our manuscript for their helpful suggestions. This work was supported by JST CREST (Grant JPMJCR1312), Japan. This research used computational resources of the K computer provided by the RIKEN Center for Computational Science (RIKEN R-CCS) through the HPCI System Research project (Project ID: hp170113 and hp170232). We thank Editage (www.editage.jp) for their English language editing service. The figures were produced using GrADS (http://cola.gmu.edu/grads/). The NCEP FNL data is available at this site (https://doi.org/10.5065/D6M043C6). The code for NICAM version 15 can be provided from the NICAM development team (http://www.nicam.jp/). The code for SCALE-RM version 5.2.5 is freely available at http://r-ccs-climate. riken.jp/scale/download/index.html under the 2-Clause BSD license. All simulation data calculated by NICAM and SCALE-RM are deposited in the local storage at RIKEN R-CCS. Data, analysis codes, and model configuration files for NICAM and SCALE-RM, used for this study, are publicly available from Zenodo repository (https:// doi.org/10.5281/zenodo.3543914).