Arakawa and Wu (2013, hereafter AW13) recently developed a formal approach to a unified parameterization of atmospheric convection for high-resolution numerical models. The work was based on ideas formulated by Arakawa et al. (2011) and was designed to parameterize the eddy transport of moist static energy by convection. AW13 contained two main points: 1) a new formulation for convective transport valid for both small and large convective cloud fractions and 2) a closure in terms of convective cloud fraction suitable for such situations. Thus, the key parameter in this approach is convective cloud fraction

In conventional parameterization, it is assumed that

*h*by convective updrafts

*c*denotes mean value in convective updrafts, a tilde denotes environmental value, and an overbar denotes average over a numerical model grid box. After simple manipulation, we express the vertical eddy transport of

*h*by convection as

*i*and

*j*represent an updraft and/or a downdraft, and the summation is over all convective updrafts and downdrafts. The convective cloud fraction includes all updraft and downdraft areas

Below we will show that (i) the assumption of

To verify this conclusion, we show in Fig. 1 the vertical transport of moisture by convective updrafts in a simulated mesoscale convective complex over the U.S. southern Great Plains using the Weather Research and Forecasting (WRF) Model at 1-km resolution (Liu et al. 2015). Note that this direct calculation bypasses the convective closure issue for determining the individual terms on the right-hand sides of Eqs. (3) and (5) in the vertical transport formulations by assuming that they are known. We divide the model domain into subdomains of different sizes to mimic the grid spacing of large-scale and mesoscale models. As in AW13, a single top-hat-type updraft is used when calculating the parameterized vertical transport. Indeed, for all resolutions the vertical transport using the conventional parameterization [Eq. (5)] and the unified parameterization [Eq. (3)] is very close, supporting our conclusion based on the formulations. The single-updraft parameterization, however, underestimates the explicitly calculated vertical eddy transport, as also noted in AW13. The underestimation is as much as 50% and is across all scales owing to the failure of a single top-hat updraft to capture the internal variability of updrafts as shown in Liu et al. (2015).

To summarize, AW13 presented a unified approach to parameterizing the vertical transport by convective updrafts for high-resolution numerical models. In their approach, updraft fraction

## Acknowledgments

This work was supported by Office of Biological and Environmental Research of the U.S. Department of Energy under Grant DE-SC0008880, the PNNL Contract DOE/PNNL 190110, and the Scientific Discovery through Advanced Computing (SciDAC) program of the U.S. Department of Energy Office of Advanced Scientific Computing Research and Office of Biological and Environmental Research. The Pacific Northwest National Laboratory (PNNL) is operated for the DOE by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830. The authors would also like to thank Dr. Yi-Chin Liu for her help in some analyses of cloud-resolving model results.

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