First, we thank the authors of Zhang et al. (2015, hereafter ZFX15) for giving us an opportunity to clarify the essential difference between the unified parameterization (Arakawa and Wu 2013, hereafter AW13) and conventional cumulus parameterizations.

The main issue of AW13 is “…reexamination of the widely used assumption that convective updrafts cover only a small fraction of a GCM grid cell. Most conventional cumulus parameterizations that explicitly use a cloud model … assume this, at least implicitly, regarding the temperature and water vapor mixing ratio carried by GCM grid points as if they represent those of the cloud environment.” If we represent the updraft fractional area by , then this assumption is that . ZFX15 state, on the other hand, that “…the assumption of was unnecessary in the conventional approach … 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.” The purpose of this reply is to point out that this last statement, which effectively denies the existence of the gray-zone problem, has no logical basis.

We begin our discussion with the following expression for the vertical eddy transport of moist static energy:

 
formula

Here the notations follow AW13 and ZFX15. In particular, the subscript c and the tilde represent values for the convective updrafts and their environment, respectively, and the overbar and the prime denote the average over the entire area of the GCM grid cell and the deviation from that average, respectively. Equation (1) is identical to Eq. (3) in ZFX15 and equivalent to Eq. (9) in AW13. Using the definition , we can easily show that

 
formula

With this relation and the approximation , Eq. (1) becomes

 
formula

which is identical to Eq. (8) in ZFX15.

The source of confusion here seems to be that ZFX15 view the assumption only in the context of justifying Eq. (3). They point out that since does not have to be assumed in the derivation, Eq. (3) is basically the same as the corresponding expression in the unified parameterization. They further claim that the “scale-awareness factor” is implicitly included in Eq. (3) through Eq. (2). Parameterization, however, means more than Eq. (3). In this equation, the gridpoint value of the moist static energy can be used for , and a plume model embedded in the grid cell can be used to calculate and , leaving and as two unknowns. (In most conventional parameterizations, and are combined as the updraft mass flux, , which replaces as one of the unknowns.) Because there are two unknowns, we need another physical idea to close the system, allowing the two unknowns to be determined simultaneously. This is where the essential difference between the unified and conventional parameterizations appears.

As Arakawa (2004) pointed out, all cumulus parameterization schemes in current use can be interpreted as adjustment schemes. The adjustment takes place primarily through the modification of the cloud environment by convection, as formulated by Arakawa and Schubert (1974). The area of the cloud environment that accommodates such modification decreases as increases and, in the limit of , no adjustment takes place within the grid cell occupied by the updraft. Thus, the assumption of small is implicit in the concept of adjustment as applied to a single-column parameterization.

As in AW13, let be the value of required for full adjustment to equilibrium in response to the grid-scale destabilization. Since depends on the grid-scale processes only, it is a known quantity as far as parameterization of the subgrid processes is concerned. For full adjustment, is needed. Using this together with the definition of M, Eq. (3) may be rewritten as

 
formula

The mass flux can then be found by dividing the left-hand side by , which is according to Eq. (2). Here it is important to note that the factor , called the scale-awareness factor by ZFX15, is used for division, not for multiplication. If this parameterization is formally applied with , for example, then the resulting M is not finite. This nonphysical result is obtained because of an inconsistency: the use of Eq. (4) with large is inconsistent with the assumption of small that is implicit in the use of full adjustment for the left-hand side. Most conventional parameterizations use a “relaxed adjustment,” in which only a constant fraction of is applied at each time step. The situation is still basically the same as in the case of full adjustment as long as the relaxation factor is independent of .

In the unified parameterization, the logical flow is almost reversed. The closure is achieved through determination of for each realization of the grid-scale processes. AW13 point out that the simplest physically reasonable choice for is

 
formula

It can easily be seen that Eq. (5) automatically satisfies the condition as long as and have the same sign. By eliminating between Eqs. (5) and (1), we obtain

 
formula

Equation (6) is identical to Eq. (15) in AW13. This is how the unified parameterization is implemented in prognostic models. Since , Eq. (6) has the form of a relaxed adjustment, but here the relaxation factor depends on and vanishes for . In a sharp contrast, most conventional parameterizations do not even attempt to determine so that no explicit information about can be used in the adjustment.

Finally, we point out that the blue and red curves in Fig. 1 of ZFX15 do not show the results that would be obtained with the conventional and unified approaches. In plotting these curves, the authors must have diagnosed the right-hand sides of Eqs. (3) and (1), including the values of , directly from data. In this way, they bypassed the closures, which are essential to the dependence of the parameterized results.

Acknowledgments

We thank Professors David Randall and Steven Krueger for their many suggestions to improve the original manuscript. The first author is supported by the National Science Foundation Science and Technology Center for Multi-Scale Modeling of Atmospheric Processes, managed by Colorado State University under Cooperative Agreement ATM-0425247. The second author is supported by Taiwan’s National Research Council through Grant 103-2111-M-002-004 to National Taiwan University.

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Footnotes

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-12-0330.1.