Dependence of Convective Cloud Properties and Their Transport on Cloud Fraction and GCM Resolution Diagnosed from a Cloud-Resolving Model Simulation

Abstract

The scale-aware convective parameterization for high resolution global climate models must satisfy the requirement that the parameterized subgrid convective transport diminishes as the model resolution increases to convection-resolving resolutions. A major assumption in current scale-aware convection schemes is that the differences between convective cloud properties and their environmental counterparts are independent of cloud fraction. This study examines convective cloud vertical velocity, moist static energy (MSE), moisture, and the vertical eddy transport of MSE and moisture for different averaging subdomain sizes and fractional convective cloudiness using a cloud resolving model simulation of a midlatitude mesoscale convective system. Results show that convective cloud fraction, mass flux, and vertical transport of MSE and moisture increase with decreasing subdomain size. The differences between convective cloud properties in both updrafts and downdrafts and their environment depend on both cloud fraction and the averaging subdomain size. For a given subdomain size, the differences increase with cloud fraction, in contrast to the assumption used in current scale-aware convection parameterization schemes. A consequence of this is that the parameterized convective eddy transport reaches maximum at a higher cloud fraction than believed in previous studies. This has implications on how fast the subgrid convective transport should diminish as GCM resolution increases.

1. Introduction

Convective clouds have horizontal sizes from sub-kilometers for isolated cumulus to tens of kilometers or larger for mesoscale convective systems. Due to differences between properties, such as temperature and moisture, inside these clouds and those in their environment, convection can transport a large amount of heat, moisture, momentum, and pollutants from the surface and lower troposphere to the upper troposphere. However, due to the small spatial scales relative to the grid size of global climate models (GCMs), which is on the order of 100 km or larger in the past, the collective effects of convection on grid-scale dynamic and thermodynamic fields are parameterized. As the computing power increases, GCM resolutions are also increasing. Many modeling centers have already started using high-resolution (around 25 km grid size) GCMs for climate research related to simulations of hurricanes and mesoscale convective systems [1,2,3,4,5]. It is likely in the next 5 to 10 years GCM grid-spacing will reduce to sub-10 km, the so-called “grey zone”, comparable to the size of individual deep convective clouds. In such situations, how to represent convection in climate models is an open question. Ideally, one would expect that as the GCM resolution approaches that of cloud-resolving models (CRMs), the parameterized convective heating and drying profiles would converge to their explicitly simulated counterparts. However, refs. [6,7] showed that this is not the case, suggesting that current convective parameterization schemes are unable to represent convective effects correctly when the GCM resolution approaches kilometer-scale.

Many studies have been devoted to making convective parameterization schemes scale-aware since then [7,8,9,10,11]. In these studies, the resolution dependence of the parameterized convective effects is represented through an explicit inclusion of convective cloud fraction, which was not used in past convective parameterization schemes. The intuitive reasoning is that for a given physical size of a convective cloud, as the GCM grid size decreases, the fractional area it occupies increases. Using an idealized CRM simulation, [8] showed that convective transport of heat and moisture mostly depends on convective cloud fraction. Mathematically, convective eddy transport can be expressed as [8,12]:

$$\overline{w\text{'}\psi \text{'}}=\sigma \left(1-\sigma \right)\left(w_c-w_e\right)\left({\psi }_c-{\psi }_e\right)$$

(1)

where $\sigma$ is convective cloud fraction, w is vertical velocity in convective clouds, and $\psi$ is a generic dynamic or thermodynamic property being transported, such as heat, moisture, moist static energy, or momentum. Subscripts c and e represent cloud-mean and environmental-mean values, respectively.

In the above-cited studies that use cloud fraction to represent GCM-resolution dependence of convective parameterization, it is implicitly assumed that the difference between clouds and their environmental properties (e.g., ${\psi }_c-{\psi }_e$) is independent of cloud fraction or GCM resolution. This has important implications for convective parameterization schemes for use in the grey zone since if it is true, convective eddy transport would reach maximum at $\sigma =0.5$ [7]. On the other hand, if $w_c-w_e$ and/or $\left({\psi }_c-{\psi }_e\right)$ also vary with cloud fraction, the parameterized eddy transport would have a more complex dependence on cloud fraction, and the suggested representation of the scale-awareness of the convection schemes could either underestimate or overestimate the subgrid convective eddy transport in the grey zone. To date, there has been no evaluation of the assumption in the literature. Therefore, the objective of this study is to examine the dependence of convective cloud properties and their transport on cloud fraction and GCM grid spacing using output from a CRM simulation of convective system. The rest of the paper is organized as follows: Section 2 describes the data and analysis method; Section 3 presents the results; Section 4 discusses the implications of the results; and Section 5 concludes the paper.

2. Simulation Data and Analysis Method

The data used in this study is from a simulation of a mesoscale convective system during the Midlatitude Continental Convective Clouds Experiment (MC3E) [13] in the Southern Great Plains (SGP) conducted by the U.S. Atmospheric Radiation Measurement (ARM) program. On 23 May 2011, a mesoscale convective system formed at 2100 UTC, passed through the ARM SGP site and dissipated around 0400 UTC May 24. A regional model simulation of the convective system using the Weather Research and Forecast (WRF) model was conducted and systematically evaluated against field observations in both the macro-state and micro-state of the convection system by [14]. They found that the model realistically simulated the convection system. For this reason, we use the same data as in their study. The WRF model used for the simulation is WRF 3.1.1, with a domain size of 561 km × 561 km at a horizontal grid spacing of 1 km. The model has 41 vertical layers from 1000 hPa to 100 hPa. The simulation starts at 0600 UTC May 23 and ends at the 0600 UTC May 24. The simulation data analyzed in the study covers the period from 2100 UTC May 23 to 0600 UTC May 24. The output is saved every 6 min. Figure 1 shows a snapshot of 500 hPa vertical velocity and surface precipitation at 0100 UTC May 24. In regions of heavy precipitation there are strong upward and downward motions.

To calculate the cloud properties and their transport by convection at different GCM resolutions, we coarse-grain the 1-km resolution data by averaging it over different subdomain sizes equivalent to corresponding GCM grid spacing. After excluding 24 km from each of the lateral boundaries, we divide the 512 km × 512 km domain into 256, 128, 64, … 4 km subdomains, respectively, representing GCM resolutions of 256 km, 128 km, etc., similar to the approach in several previous studies [8,15,16,17]. Thus, for a 512 km × 512 km domain at a given model output time, there are 4, 4², 4³, 4⁴, 4⁵, 4⁶, 4⁷ subdomains for 256, 128, 64, 32, 16, 8 and 4 km subdomain sizes, respectively. Hereafter, we will use “subdomain size” and “GCM resolution” interchangeably. Within each subdomain, a convective updraft grid point is identified by the following criteria: (1) vertical velocity w > 1 m/s and total hydrometeor mixing ratio ${\mathrm{q}}_{\mathrm{t}}>1\times {10}^{-6}$ kg/kg or (2) w > 2 m/s. Similarly, a convective downdraft is determined by (1) w < −1 m/s and ${\mathrm{q}}_{\mathrm{t}}>1\times {10}^{-6}$ kg/kg or (2) w < −2 m/s. Each subdomain can contain any of updrafts, downdrafts, and environment. In [8] and subsequent applications of their scale-aware convective parameterization approach [9,10,11], only updrafts are considered for convective cloud fraction and associated transport. However, convective downdrafts play an important role in the interaction between convection and its large-scale environment [18,19,20,21]. Therefore, in this study we will examine cloud properties and their transport by first considering updrafts only and then considering both updrafts and downdrafts.

The vertical eddy transport of a generic variable $\psi$ is given by

$$\overline{w\text{'}\psi \text{'}}=\overline{w\psi }-\overline{w}\overline{\psi }$$

(2)

where overbar represents subdomain mean and prime is the deviation at a grid point from the mean. The eddy transport from the direct grid-point calculation is given by

$$T_{dir}={\left(\overline{w\text{'}\psi \text{'}}\right)}_{dir}=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N\left(w_i-\overline{w}\right)\left({\psi }_i-\overline{\psi }\right)$$

(3)

where subscript i represents each grid point in the subdomain, and N is the total number of grid points in the subdomain. By definition, a subdomain mean can be expressed as

$$\overline{\psi }={\sigma }_u{\psi }_u+\left(1-{\sigma }_u\right){\psi }_e$$

(4)

where subscript u denotes updraft, subscript e denotes environment, and ${\sigma }_u$ is the updraft cloud fraction. The updraft-mean and environmental-mean values are computed over all updraft grid points and environmental grid points, respectively. For convective parameterization, one typically ignores the internal heterogeneity within updrafts and differences of properties between updrafts, as well as inhomogeneity in the environment. With this, the parameterized eddy transport within a subdomain can be written as

$$T_{par}={\left(\overline{w\text{'}\psi \text{'}}\right)}_{par}={\sigma }_u\left(1-{\sigma }_u\right)\left(w_u-w_e\right)\left({\psi }_u-{\psi }_e\right)$$

(5)

Thus, the difference between $T_{dir}$ and $T_{par}$ measures the degree of inhomogeneity within and between clouds in a cloud population as well as inhomogeneity in the convection environment. As we will show later, $T_{par}$ seriously underestimates the eddy transport from the direct calculation $T_{dir}$ for moist static energy (MSE). As shown in [15], when three groups of updrafts are considered based on their intensity, together with the consideration of downdrafts, the parameterized eddy transport is very close to that from direct calculation.

To determine the cloud properties and their transport as functions of cloud fraction and subdomain size, we will first calculate the mean in-cloud properties and the corresponding cloud fraction within each subdomain of the given size. Then, we will do an ensemble average of these properties over subdomains that have convection and leave out the convection-free subdomains.

3. Results

3.1. Dependence on Subdomain Size

Using the analysis method described above, we calculate the average cloud fraction and convective mass flux for each subdomain size, which are shown in Figure 2. As can be seen, both cloud fraction and mass flux averaged over all convective subdomains increase with decreasing subdomain size. If downdrafts are considered in the calculation of cloud fraction, the cloud fraction increases significantly for all subdomain sizes, especially for large subdomain sizes (e.g., 64 km and larger), although downdraft mass flux is relatively small, about 20 to 25%, compared to updraft mass flux. The average cloud fraction over all convective subdomains is greater than 10% at subdomain sizes of 64 km or smaller, reaching as large as 50% at subdomain size d = 4 km. This indicates that the assumption of cloud fraction $\sigma \ll 1$ used in conventional convective parameterization schemes is no longer valid for subdomain sizes smaller than 64 km.

The total and eddy vertical transport of MSE ($h=C_{p}T+gz+Lq)$ and specific humidity q are calculated for each convective subdomain and averaged for different subdomain sizes (Figure 3). Both increase with decreasing subdomain size. This is because more convection-free subdomains are removed from the averaging samples as the subdomain size decreases since only convective subdomains are counted. In terms of the vertical distribution, the total transport reaches a maximum near 500 hPa and the eddy transport reaches a maximum near 600 hPa. For large subdomains (e.g., d = 128 km) the total vertical transport is almost entirely from eddy transport, whereas for small subdomain sizes eddy transport contributes 50% (for d = 8 km) to 30% (for d = 4 km) to the total vertical transport. Similar results were found in previous studies [8,15]. The fact that the relative contribution from subgrid eddy transport to the total transport decreases with decreasing subdomain size simply means that more transport is resolved by the GCM and the importance of subgrid eddy transport diminishes as the GCM resolution increases. The total moisture transport reaches a maximum at a lower level, around 650 hPa while maximum eddy moisture transport is at about 600 hPa. For both eddy and total fluxes, moisture transport contributes more than 70% to the MSE transport.

3.2. Cloud Fraction Dependence without Considering Downdrafts

From Equation (5), the eddy transport at different GCM resolutions can be expressed as a function of cloud fraction in the form of ${\sigma }_u\left(1-{\sigma }_u\right)$ if $\left(w_u-w_e\right)\left({\psi }_u-{\psi }_e\right)$ is independent of cloud fraction. If this is true, it will make building scale-aware convection parameterization relatively simpler. One can parameterize the cloud fraction and determine $\left(w_u-w_e\right)\left({\psi }_u-{\psi }_e\right)$ from cloud models. Indeed, this is what [8] suggested. Since this assumption has huge implications for scale-aware convection parameterization development, in this section we will examine whether the cloud properties and their difference with the corresponding environmental values depend on cloud fraction.

Figure 4 shows the vertical velocity in updrafts, the environment, and their difference as functions of cloud fraction and subdomain size. There is a clear dependence of all three on both cloud fraction and subdomain size. The mean updraft vertical velocity generally increases with cloud fraction and decreases with decreasing subdomain size. This is unexpected from a single cloud perspective. When there is a convective cloud in a model subdomain, reducing the subdomain size that contains the cloud should not affect the updraft intensity. Similarly, as the subdomain size decreases, while the fractional area occupied by the cloud increases, it should not affect the average vertical velocity within the cloud either. However, statistically when there is a population of many convective clouds in a model domain and the domain is divided into many subdomains of a given size, smaller cloud fraction in a subdomain is likely due to either small clouds or cloud edges being cut through by the dividing grid lines of the subdomain. Therefore, the mean vertical velocity in these small clouds or partial clouds is small. Conversely, a larger cloud fraction likely corresponds to larger clouds in the subdomain, which are generally more intense, resulting in the increase in mean updraft velocity with cloud fraction. Likewise, a given cloud fraction in a larger subdomain means more large clouds, thus having stronger mean upward vertical velocity, and vice versa.

The environmental vertical velocity is generally small and negative. It has less cloud fraction and subdomain size dependence. Two factors can explain the negative environmental vertical velocity. First, because of strong upward motion in updrafts, there is compensatory downward motion in the convection environment. Second, when downdrafts are not considered a part of convective cloud, they are included in the environment and thus contribute to the environmental vertical velocity. This is especially true for moderately large cloud fraction (0.4 to 0.8) and medium-sized subdomains (64 to 8 km). The cloud fraction and subdomain size dependencies of the difference between updraft and environmental vertical velocities essentially follow those of the updraft vertical velocity due to relatively small values of $w_e$.

Figure 5 shows the cloud-mean, environment-mean, and their differences of MSE and specific humidity as functions of cloud fraction and subdomain size at 500 hPa. Since MSE is on the order of ~300 K and their cloud-environment difference is only a few degrees, a model-domain and simulation-time-mean is subtracted from the actual $h_u$, $h_e$, $q_u$ and $q_e$ values for display purposes. Similar to vertical velocity, there is a strong dependence of $h_u$ and $q_u$ on both cloud fraction and subdomain size. Larger clouds have less entrainment dilution, thus higher MSE and specific humidity. Unlike vertical velocity, the environmental values of MSE and moisture are only slightly smaller than their updraft counterparts and are also cloud fraction and subdomain size dependent. The cloud-environment differences of MSE also show clear dependence on cloud fraction and subdomain size. The cloud-environment differences of moisture depend more on subdomain size than on cloud fraction. For a given subdomain size, updrafts are generally more buoyant and moister than their environment when cloud fraction is larger although there are some spotty exceptions. For a given cloud fraction, updrafts in larger subdomains are more buoyant and moister than their environment.

Ref. [8] suggested that compared to $\left(w_u-w_e\right)$ and $\left(h_u-h_e\right)$ individually, their product $\left(w_u-w_e\right)\left(h_u-h_e\right)$ can be treated as independent of cloud fraction. Figure 6 shows that, similar to the difference of vertical velocity and MSE between the updrafts and their environment, their product also has strong dependencies on cloud fraction and subdomain size. The same is for humidity. This makes the parameterization of eddy transport much more complicated than has been assumed in existing scale-aware convection parameterization schemes (e.g., [10,11]). In addition to the factor of ${\sigma }_u\left(1-{\sigma }_u\right)$, one needs to account for the dependence of $\left(w_u-w_e\right)\left(h_u-h_e\right)$ on cloud fraction and GCM resolution for convective transport of MSE. The same is for moisture transport.

Finally, we show in Figure 7 the eddy transport of MSE directly calculated using Equation (3) and that from what the parameterization from Equation (5) would give if cloud properties and their differences with their environment were known. Both have strong dependence on cloud fraction and subdomain size. However, there are important differences. First, the MSE flux from direct calculation reaches the maximum at cloud fraction greater than 0.8 for small subdomain sizes. On the other hand, the parameterized MSE flux reaches the maximum when the cloud fraction is in the range of 0.5 to 0.8, partly due to the factor of ${\sigma }_u\left(1-{\sigma }_u\right)$, which reaches the maximum at cloud fraction of 0.5. Note that in obtaining Figure 7b, the differences between the cloud properties and their environmental properties (e.g., $w_u-w_e$ and $h_u-h_e$) were not assumed to be independent of cloud fraction, but rather calculated directly from the model output. Second, the parametrized MSE flux is much smaller than that from direct calculation, representing only about 30 to 50%. This indicates that there are large variations of cloud properties either within updrafts or between updrafts so that even when convective updrafts nearly occupy the entire subdomain the internal eddy correlation still contributes a large amount of eddy flux. This is consistent with [15], who showed that using a single top-hat updraft model is not accurate enough for convective transport estimation, and three updraft types grouped by their vertical velocity and one downdraft type are needed to represent convective transport more accurately.

3.3. Cloud Fraction Dependence Considering Downdrafts

Convective downdrafts play an important role in stabilizing the atmosphere. Although mass flux in downdrafts is less than updrafts, downdrafts are an important part of convection development and transport processes, especially in organized convective systems [22,23,24,25]. Most of convective parameterization schemes include the effect of downdrafts [18,26,27,28]. In the last subsection, convective cloud fraction only included updrafts, same as in existing scale-aware convective parameterization development. When both updrafts and downdrafts are considered, the eddy transport can be written as

$$\overline{w^{\prime }{\psi }^{\prime }}={\sigma }_u\left(1-{\sigma }_u\right)\left(w_u-w_e\right)\left({\psi }_u-{\psi }_e\right)+{\sigma }_d\left(1-{\sigma }_d\right)\left(w_d-w_e\right)\left({\psi }_d-{\psi }_e\right)$$

(6)

$$-{\sigma }_u{\sigma }_d\left[\left(w_u-w_e\right)\left({\psi }_d-{\psi }_e\right)+\left(w_d-w_e\right)\left({\psi }_u-{\psi }_e\right)\right]$$

The first line is contributions from updrafts and downdrafts separately and the second line is from their interaction. In addition to the differences between updraft and environmental properties, we also need to know the differences between downdraft and environmental properties. Note that when downdrafts are considered as a part of convection, the fractional area of the convection-free environment is determined by $1-{\sigma }_u-{\sigma }_d$ instead of just $1-{\sigma }_u$ as in the last subsection. We found that although the environmental values are modified due to the exclusion of downdrafts from the definition of “environment” in this subsection, the differences between updraft properties and their environmental values and their dependence on cloud fraction and subdomain size are very close to those shown in the last subsection. Thus, we will only show the downdraft properties in this subsection.

Figure 8 shows the relevant quantities for vertical velocity. Note that the downdraft properties and their differences with the environment are meaningful only when there are downdrafts. Thus, they are plotted as functions of downdraft cloud fraction and subdomain size. Several points can be made. First, downdraft vertical velocity can be as large as −3 m/s and is also dependent on downdraft cloud fraction and subdomain size. Second, the environmental vertical velocity is still negative, but very small in magnitude. As a result, the differences between updraft and environmental vertical velocities $w_u-w_e$ are smaller than without considering downdrafts (cf. Figure 4c). Third, compared to $w_u-w_e$, $w_d-w_e$ is less dependent on cloud fraction and subdomain size, although it still shows similar patterns of dependencies.

Figure 9 is the same as Figure 5 except for downdraft MSE and moisture. Again, there is strong dependence of MSE in downdrafts and the environment on downdraft cloud fraction. From $h_d-h_e$, positive values are in the lower left part and negative values are in the upper right part of the plot. These indicate that smaller downdrafts are positively buoyant and larger downdrafts are negatively buoyant. This pattern is different from that of $h_u-h_e$ in Figure 5. From Figure 3, downdraft mass flux reaches a local maximum near 500 hPa and becomes smaller below it. One probable explanation is that many of the small downdrafts are forced downdrafts and are positively buoyant. As they move downward, they decelerate and detrain out. Only relatively larger downdrafts can survive the entrainment and are negatively buoyant. Similar to MSE, moisture in downdrafts have moisture excess for small downdrafts and moisture deficit for large downdrafts.

Figure 10 shows $\left(w_d-w_e\right)\left(h_d-h_e\right)$, $\left(w_d-w_e\right)\left(q_d-q_e\right)$ and parameterized eddy transport of MSE from Equation (6). Compared to Figure 5, the cloud fraction and subdomain size dependences of $\left(w_d-w_e\right)\left(h_d-h_e\right)$ and $\left(w_d-w_e\right)\left(q_d-q_e\right)$ are minimal except for very small cloud fractions. Thus, it is safe to assume that these two quantities are independent of cloud fraction and GCM resolution for parameterizing convective eddy transport in the grey zone. The parameterized eddy transport after including downdrafts as functions of cloud fraction and subdomain size (Figure 10c) is in better agreement with that from direct calculation, particularly in its pattern of dependence on cloud fraction and subdomain size, although the magnitudes are still significantly smaller.

4. Discussion

Having shown that the cloud-environment differences of vertical velocity, MSE and moisture increase with both updraft and downdraft cloud fractions for a given subdomain size, in this section, we discuss their implications for parameterization of subgrid transport. For simplicity, we will ignore downdrafts. They can be easily incorporated similar to Section 3.3.

Based on the results shown in the last section, we can assume to first order approximation that the cloud-environment differences are proportional to cloud fraction, that is,

$$w_u-w_e~{\sigma }_u{\left(w_u-w_e\right)}^{*}$$

(7)

$$h_u-h_e~{\sigma }_u{\left(h_u-h_e\right)}^{*}$$

(8)

$$q_u-q_e~{\sigma }_u{\left(q_u-q_e\right)}^{*}$$

(9)

where the asterisks denote quantities that are independent of cloud fraction. With this, Equation (5) for a generic thermodynamic variable can be rewritten as:

$$T_{par}={\left(\overline{w\text{'}\psi \text{'}}\right)}_{par}={\sigma }_u^3\left(1-{\sigma }_u\right){(w_u-w_e)}^{*}{\left({\psi }_u-{\psi }_e\right)}^{*}$$

(10)

It reaches a maximum at ${\sigma }_u=0.75$. This is consistent with the observation that the parameterized eddy transport has a broad maximum in the cloud fraction range of 0.6 to 0.8 in Figure 7b. Ref. [8] assumed that the cloud-environment differences are independent of cloud fraction. As a result, the parameterized eddy transport would reach a maximum at ${\sigma }_u=0.5$. This differs significantly from Figure 7b. One possible reason for the difference is that convection is unorganized whereas convection in our case is a highly organized mesoscale convective system.

5. Conclusions

Using a cloud-resolving model simulation of a midlatitude convective system with the WRF model, by coarse-graining the high-resolution data to averages over subdomains whose sizes are equivalent to various GCM resolutions, this study shows that both convective cloud fraction and mass flux increase with decreasing subdomain size. This is because more convection-free subdomains are removed from the averaging process as the subdomain size decreases. Both subgrid scale convective eddy transport and the total transport of moist static energy and moisture also increase as the subdomain size decreases. However, the relative contribution from the eddy transport to the total transport decreases as the subdomain size decreases, as also noted in previous studies [8,15].

Developing convection parameterization schemes for global climate models that are aware of such GCM resolution dependence as the model resolutions increase to the grey zone has been an active research topic in recent years. A major assumption is that the differences between cloud properties and their environmental counterparts are independent of cloud fraction and model resolution. This study examines whether the cloud-environment differences of vertical velocity, moist static energy and specific humidity are independent of cloud fraction and model resolution. We showed that, to the contrary of the assumption, there is strong dependence of these fields on both cloud fraction and GCM resolution. The updraft vertical velocity increases with cloud fraction for a given GCM resolution. For a given cloud fraction, it decreases with GCM resolution. The dependence of the difference between updraft and its environmental vertical velocity on cloud fraction and GCM resolution largely follows that of the updraft vertical velocity. There is also a strong cloud fraction and GCM resolution dependence of the differences in moist static energy and moisture between the updrafts and their environment. An explanation for these is that larger updrafts are generally more intense, more buoyant, probably due to less entrainment and thus moister.

When downdrafts are considered in convective transport parameterization, we also need to know the dependence of the differences between downdrafts and their environment of vertical velocity, MSE and moisture on downdraft area fraction and subdomain size. It is found that, like their updraft counterpart, downdraft properties also have strong dependence on cloud fraction and subdomain size. Downdraft air with larger area fraction in smaller subdomains are negatively buoyant and drier than their environment. Conversely, downdrafts with smaller area fraction in larger subdomains are positively buoyant and moister than their environment. However, the dependence of the products of vertical velocity difference and moist static energy (or moisture) difference on downdraft cloud fraction and GCM resolution is very weak and can be safely ignored for parameterization purposes.

The results presented in this study are for a well-organized mesoscale convective system. Whether they also hold for scattered, unorganized convection is not clear although we doubt the qualitative conclusions will change. Thus, it is desirable to carry out similar analyses for unorganized convection. This study uses the WRF model simulation data. It is possible that the results also depend on the model we use. However, in our opinion, this is not probable as long as a model simulates a convective system realistically. Nonetheless, it would be helpful if the results are confirmed with a different model in the future.