## 1. Introduction

The wake parameterization proposed by Grandpeix and Lafore (2010, hereafter GL) can be considered as a special application of the general mode decomposition approach proposed by Yano et al. (2005). According to the latter work, the mass flux–based parameterization, originally introduced by Arakawa and Schubert (1974, hereafter AS74), can be generalized by considering it as an approach based on segmentally constant mode decomposition [or segmentally constant approximation (SCA); Yano et al. 2010]. The idea of SCA consists of subdividing a gridbox domain into a number of constant-value segments in different sizes and shapes, each representing various subgrid-scale subcomponents, not only the convective elements but also elements such as the wake. In this respect, GL’s wake parameterization is a particular case of generalization of the mass flux convection parameterization based on SCA.

Here, it is emphasized that the formulation in AS74 is already presented in a very general manner by retaining an effect of finite fractional area *σ _{j}* occupied by each cloud type, and an asymptotic limit,

*σ*→ 0, is taken only at the last stage of their paper. It is also emphasized that possibilities of interpreting AS74’s mass flux formulation from a more general perspective are already clearly stated in Yano et al. (2005).

_{j}Unfortunately, GL developed their parameterization on their own without referring to either Yano et al. (2005) or AS74. This led GL to various mistakes and ambiguous assumptions. Many equations are presented without derivations. This paper lists corrections based on rigorous application of SCA. The list intends to demonstrate the close link of their parameterization to the mass flux formulation as well as the importance of developing a parameterization from first principles as emphasized by Yano et al. (2005). The latter point is extremely important because, as it stands, it is totally unclear under what principle and with what kind of statistical and probabilistic approaches GL have developed their wake parameterization. Some general remarks are added wherever possible for this reason but are kept to a minimum for brevity.

## 2. Corrections and comments

The following corrections and comments are in the order of appearance in GL with GL’s section numbers referenced in the headings. To present these corrections efficiently in the text, mathematical derivations are provided separately in the appendix. By following the logical development of the SCA formulation, derivations are given in a different order in the appendix.

All equations numbered without prefixes are those from GL, unless otherwise noted. Our equations in the main text and the appendix are numbered with prefixes 2 and A, respectively, for distinction. Definitions of mathematical symbols are as in GL unless explicitly defined in the text.

### a. Comments on GL’s section 3a

**r**

_{Γ,w}designating the position of the wake boundary. This is obtained directly from Eq. (A.6) by setting

*j*=

*w*and

*x*. Their Eq. (7) is found only if the wake boundaries are perfectly vertical. This assumption is never explicitly stated in GL.

The flag *η* is introduced below Eq. (7): it is against the principle of scale separation to consider the case in which all wakes are clustered around the center of a grid box so that they do not cross the gridbox boundary. It is not clear how wakes can physically know the existence of a gridbox boundary. For consistency with scale separation, the wakes should distribute smoothly over the grid boxes. For this reason, *η* = 1 is the only consistent choice. See section 3.2 of Yano (2009) for further discussion on the principle of scale separation.

Here, *j* = *w*,

The original Eq. (8) is found only if we assume that (i) *σ _{w}* is constant and independent of large-scale (gridscale) coordinates. The first assumption is already questionable because there is no reason to expect that the whole wake flows

*σ*for the wakes is calculated at each grid box prognostically by their Eq. (10). Note that the error in Eq. (8) is inherited in their subsequent development with Eqs. (9), (10), (11), and (12).

_{w}In their definitions of the entrainment and detrainment rates, *e _{w}* and

*d*, given just before Eq. (10),

_{w}**V**should be replaced by

**V*** [cf. Eq. (2.2)]. Note that this is a direct consequence of a mistake found in their Eq. (7).

### b. Comments on GL’s section 3b

We should set *δω*^{cv} = −*g*[*M _{p}*/

*σ*− (

_{w}*M*+

_{c}*M*)/(1 −

_{e}*σ*)] rather than

_{w}*δω*

^{cv}= −

*g*[

*M*/

_{p}*σ*−

_{w}*M*/(1 −

_{c}*σ*)] because the nonwake area is occupied by both the convection and the environmental subcomponents.

_{w}### c. Comments on GL’s section 3c

Note that in AS74, convective elements are assumed not to cross the gridbox boundary; thus, the last term in the right-hand side is missing. This assumption is valid in an asymptotic limit to *σ _{w}* → 0 for the wake component. On the other hand, in this limit, the last term in Eq. (2.5b) reduces to the gridbox mean divergence. Equation (A.5) with

*φ*=

*θ*should be used for further reduction of these last terms in both Eqs. (2.5a) and (2.5b) subsequently.

Regarding assumption (ii) immediately below Eq. (14), it is wrong to replace *θ* and **V**, respectively, by *j* = *w* and *x* in the contour integrals over **V**. It is a mistake to neglect the fact that all of the subgrid-scale variables, including

Note that the advection by wake and off-wake flows is included in Eqs. (2.6a) and (2.6b) as a result of the correction.

If the paired Eqs. (18) are derived correctly, the first of the pair should correspond to Eqs. (44) and (48) of AS74 and the second to Eq. (16) of AS74, when the static energy in the latter is redefined as potential temperature. However, note that *η* = 0 is implicitly assumed in AS74.

There is no possible way for obtaining the so-called gravity wave radiation term in the second of the paired Eqs. (19) when assumption (iii) immediately below Eq. (14), corresponding to the upstream approximation, is strictly applied. Here, lack of justification should not be obscured by an expression such as “heuristic representation.”

*θ*does not strictly follow an upstream approximation, but rather represents a finite deviation from this state designated by

*j*=

*w*and

*x*, respectively, and

*σ*= 1 −

_{x}*σ*. A closed expression for this term is obtained by applying the eddy–diffusion hypothesis introduced by, for instance, Asai and Kasahara (1967) with an eddy–diffusion coefficient given by

_{w}*μ*. Then we add the terms −(

_{e}*μ*/

_{e}*σ*)

_{w}*δθ*and

*μ*/(1 −

_{e}*σ*)

_{w}*δθ*, respectively, to the right-hand side of the first and the second parts of Eqs. (18). This furthermore adds a term −

*μ*[1/

_{e}*σ*+ 1/(1 −

_{w}*σ*)]

_{w}*δθ*to the right-hand side of the second in the pair of Eqs. (19), or Eq. (2.7b). GL’s original expression is found by setting

*k*

_{GW}/

*τ*

_{GW}=

*μ*[1/

_{e}*σ*+ 1/(1 −

_{w}*σ*)].

_{w}As this derivation suggests, this damping term has nothing particularly to do with the gravity waves but is simply a sum of all transport by “eddies” (including turbulent mixing) crossing the wake boundary. The most important point is that unless fluctuation *e _{w}* and

*d*to

_{w}*e*+

_{w}*μ*and

_{e}*d*+

_{w}*μ*, respectively (cf. de Rooy and Siebesma 2010).

_{e}### d. Comments on GL’s section 4c(1)

The authors should ask a serious question why a triggering condition is required for this problem (cf. Yano 2011).

### e. Comments on GL’s section 4c(2)iii

Equation (31) is derived in a highly heuristic manner and it is hard to follow. The difficulty partially stems from the fact that one of the key equations [the unnumbered equation immediately after Eq. (30)] is stated without derivation. The same result is even difficult to obtain by rigorously applying SCA.

*z*=

*z*to

_{A}*z*leads to

_{B}*K*at the updraft boundary are respectively defined by

*p*′ designates the pressure deviation from the average over the off-wake region at the given vertical level.

**V**

_{Γ}−

**V*** =

**C**

_{*}(in the Doppler-shifted sense)—and we set

**C**

_{*}·

**n**

*=*

_{u}*C*

_{*}. By furthermore setting

*p*′ ≃ 0,

*ε*≃ 0, 〈

_{u}*w*〉

*≃ 0, 〈*

_{A}*w*″

^{2}〉

*≃ 0, we finally obtain*

_{A}*M*≃

*ρσ*〈

_{B}*w*〉

*, but it does not become identical.*

_{B}## Acknowledgments

The present work is performed under a framework of the COST Action ES0905. Careful reading of the text by M. Freer is acknowledged.

## APPENDIX

### Mathematical Details

In the appendix, we consider general subgrid-scale processes consisting of an undefined number of subcomponents. Each subcomponent is designated by a subscript *j*. We refer to section 4 of Yano et al. (2005) and section 3 of Yano et al. (2010) for general discussions of the SCA system. The appendix of Siebesma (1998) would also be helpful. The notations are the same as in GL except for the general subscript *j* that is adopted here.

#### a. Derivation of a full NAM–SCA system

The nonhydrostatic-anelastic model under segmentally constant approximation (NAM–SCA) system is derived as Eq. (4.1) in Yano et al. (2005) when the boundary Γ* _{j}* separating the

*j*th subcomponent from the other subcomponents does not move with time. A two-dimensional case with a moving subcomponent–boundary is given by Eq. (3.4) of Yano et al. (2010). This section presents the formulation of a full NAM–SCA system by generalizing these results.

*φ*and assume that it is governed by

*F*is forcing (or source) for the given variable. For example, for the potential temperature

*θ*as discussed in section 3 of GL, forcing is given by

*F*=

*θH*/

*TC*.

_{p}*φ*for a

*j*th subcomponent of subgrid-scale variability is obtained by integrating Eq. (A.1) over an area Σ

*occupied by the*

_{j}*j*th subcomponent:

*j*′ designates the subcomponent immediately surrounding the

*j*th subcomponent. The result Eq. (A.3) can furthermore be substituted into Eq. (A.2), which gives

#### b. Contributions from the gridbox boundary

*L*is the length of the side of the grid box. We divide the contour integral into four segments as contributions (in clockwise order) from the sides at

*x*=

*x*+

*L*/2,

*y*=

*y*−

*L*/2,

*x*=

*x*−

*L*/2, and

*y*=

*y*+

*L*/2, designated by

In the above derivation, we note that *σ _{j}* are all slowly varying variables on large-scale (gridbox) coordinates [cf. the multiscale analysis by Majda (2007)]. At the final step, we approximate the finite difference over the gridbox size

*L*by a large-scale (grid scale) divergence. Average values along the side of the boundary for

**V**

*,*

_{j}*σ*are assumed in the two middle expressions, indicating only the relevant coordinate in concern.

_{j}#### c. Mass budget equation

#### d. Velocity equations

In preparation for deriving the kinetic energy budget in the next section, this section presents the velocity equations under SCA. From this section, we switch to the geometrical vertical coordinate *z* with the vertical velocity given by *w*, being consistent with section 4 of GL.

*ρ*and

*θ*

_{0}are reference vertical profiles for the density and the potential temperature, respectively. Here,

*θ*′ designates a deviation of the potential temperature from the reference

*θ*

_{0}. The subscript

*υ*for indicating the virtual effect is omitted in the appendix in order to simplify the expressions.

*φ*in section A of the appendix, we find an expression for the vertical velocity equation under SCA for the

*j*th subcomponent:

*j*=

*w*and

*x*for the wake problem. The contour integral in the left-hand side can be rewritten in an analogous manner to Eq. (A.3).

**V**can also be introduced under SCA. The derivation is analogous to that of the vertical velocity, and it is given by

#### e. Kinetic energy budget

The kinetic energy budget equation for the *j*th subcomponent is obtained by multiplying

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