Reply to “Comments on ‘A Unified Representation of Deep Moist Convection in Numerical Modeling of the Atmosphere. Part I’”

Abstract

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 byArakawaetal.(2011)andwasdesignedtoparameterize 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 convectivecloudfractionsuitableforsuchsituations.Thus, the key parameter in this approach is convective cloud fraction s. Since only updrafts were considered in AW13, convective cloud fraction was synonymous to updraft fraction. To avoid any confusion, we will use ‘‘updraft fraction’’ exclusively in this note to mean convective cloud fraction in AW13, unless explicitly stated otherwise. In conventional parameterization, it is assumed that s � 1. This assumption is no longer valid when the horizontal resolution of numerical models approaches a few to a few tens of kilometers, since in such situations updraft fraction can be comparable to unity. Therefore, AW13 argue that the conventional approach to formulating an expression for convective transport must include a factor 12s in order to unify the parameterization for the full range of model resolutions so that it is scale aware and valid for large updraft fractions. AW13 raised an important issue in future convective parameterization development. In this note we intend to show that the assumptionofs � 1 was unnecessaryintheconventional approachandthescale-awarenessfactor12s wasalready built in implicitly, although not recognized for the last 40 years. Therefore, the formulation for convective transport in conventional parameterizations should work well even in the gray zone where large updraft fractions exist. To begin, we use the same notation as in AW13 .T o parameterize the vertical transport of moist static energy h by convective updrafts (w 0 h 0 ), we note that any variable c can be written as