• Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674701.

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  • Asai, T., and A. Kasahara, 1967: A theoretical study of the compensating downdraft motions associated with cumulus clouds. J. Atmos. Sci., 24, 487496.

    • Search Google Scholar
    • Export Citation
  • de Rooy, W. C., and A. P. Siebesma, 2010: Analytical expressions for entrainment and detrainment in cumulus convection. Quart. J. Roy. Meteor. Soc., 136, 12161227, doi:10.1002/qj.640.

    • Search Google Scholar
    • Export Citation
  • Grandpeix, J.-Y., and J.-P. Lafore, 2010: A density current parameterization coupled with Emanuel’s convection scheme. Part I: The models. J. Atmos. Sci., 67, 881897.

    • Search Google Scholar
    • Export Citation
  • Gyarmati, I., 1970: Non-Equilibrium Thermodynamics. Springer, 184 pp.

  • Majda, A. J., 2007: New multiscale models and self-similarity in tropical convection. J. Atmos. Sci., 64, 13931404.

  • Siebesma, A. P., 1998: Shallow cumulus convection. Buoyancy Convection in Geophysical Flows, E. J. Plate et al., Eds., Kluwer Academic, 441–486.

  • Yano, J.-I., 2009: Deep-convective vertical transport: What is mass flux? Atmos. Chem. Phys. Discuss., 9, 35353553.

  • Yano, J.-I., 2011: Interactive comment on “Simulating deep convection with a shallow convection scheme” by C. Hohenegger and C. S. Bretherton. Atmos. Chem. Phys. Discuss., 11, C2411–C2425.

  • Yano, J.-I., J.-L. Redelsperger, F. Guichard, and P. Bechtold, 2005: Mode decomposition as a methodology for developing convective-scale representations in global models. Quart. J. Roy. Meteor. Soc., 131, 23132336.

    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., P. Benard, F. Couvreux, and A. Lahellec, 2010: NAM-SCA: A nonhydrostatic anelastic model with segmentally constant approximations. Mon. Wea. Rev., 138, 19571974.

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    • Export Citation
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Comments on “A Density Current Parameterization Coupled with Emanuel’s Convection Scheme. Part I: The Models”

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  • 1 GAME/CNRM, Météo-France and CNRS, Toulouse, France
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Corresponding author address: Jun-Ichi Yano, GNRM, Météo-France, 42 av Coriolis, 31057 Toulouse CEDEX, France. E-mail: jun-ichi.yano@meteo.fr

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/2009JAS3044.1.

Corresponding author address: Jun-Ichi Yano, GNRM, Météo-France, 42 av Coriolis, 31057 Toulouse CEDEX, France. E-mail: jun-ichi.yano@meteo.fr

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/2009JAS3044.1.

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 σj occupied by each cloud type, and an asymptotic limit, σj → 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).

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

Equation (7), presented without derivation, should read
e2.1a
e2.1b
where
e2.2
with rΓ,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.

Equation (8) should read as follows:
e2.3

Here, is the velocity averaged over the wake regions, which is a slowly varying function over the grid boxes. The result is obtained by setting j = w, in Eq. (A.5).

The original Eq. (8) is found only if we assume that (i) and (ii) σw 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 move in the same manner as the whole gridbox mean . Here, an analogy of the wake dynamics with a colloidal fluid (multicomponent flow) may be noted (cf. Gyarmati 1970). The second assumption contradicts the fact that a fractional area σw 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).

In their definitions of the entrainment and detrainment rates, ew and dw, given just before Eq. (10), 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).

Equation (10) should read
e2.4
as derived in section C of the appendix. This equation is comparable to Eqs. (43) and (47) of AS74. Note that in AS74 the possibility of convective elements crossing the gridbox boundary is excluded, so the last term in Eq. (2.4) is missing. Both Eqs. (11) and (12) must also be modified accordingly.

b. Comments on GL’s section 3b

We should set δωcv = −g[Mp/σw − (Mc + Me)/(1 − σw)] rather than δωcv = −g[Mp/σwMc/(1 − σw)] because the nonwake area is occupied by both the convection and the environmental subcomponents.

c. Comments on GL’s section 3c

Equation (13) should read
e2.5a
e2.5b
as a special case of Eq. (A.4). These equations are a generalization of Eqs. (14), (15), (44)–(46), and (48)–(50) of AS74.

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 σw → 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 and , averaged over the given fixed grid box with j = w and x in the contour integrals over . They even fail to mention the assumption concerning V. It is a mistake to neglect the fact that all of the subgrid-scale variables, including and , are varying over the large scale (gridscale). This mistake culminates as the disappearance of the large-scale (gridscale) horizontal advection in their final result, Eq. (19).

Equation (16) should read
e2.6a
e2.6b

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.

Equation (19) should read
e2.7a
e2.7b
after corrections.

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.”

An obvious way for getting the gravity wave radiation term is to assume instead that the wake boundary value for θ does not strictly follow an upstream approximation, but rather represents a finite deviation from this state designated by . As a result, we find an additional term
e2.8
on the right-hand side of both Eqs. (18) with j = w and x, respectively, and σx = 1 − σw. 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 μe. Then we add the terms −(μe/σw)δθ and μe/(1 − σw)δθ, respectively, to the right-hand side of the first and the second parts of Eqs. (18). This furthermore adds a term −μe[1/σw + 1/(1 − σw)]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 kGW/τGW = μe[1/σw + 1/(1 − σw)].

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 along the boundary is assumed, no additional damping term arises. In other words, without this additional assumption, gravity waves that may well exist outside the wake have absolutely no effect on wake. We should also realize that addition of this term is inconsequential because this is simply equivalent to changing the entrainment and detrainment rates from ew and dw to ew + μe and dw + μe, respectively (cf. de Rooy and Siebesma 2010).

The first of the paired Eqs. (20) should read
e2.9
by correctly performing the derivation of Eqs. (19) from Eqs. (17) and (18). Readers are strongly encouraged to verify this equation by themselves.

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.

An exact kinetic energy budget of the SCA system is given by Eq. (A.11). Its vertical integral (assuming steadiness) over the updraft region from the height z = zA to zB leads to
fd2.10
where the updraft kinetic energy and the kinetic energy K at the updraft boundary are respectively defined by
e2.11a
e2.11b
Various source terms on the right-hand side are defined by
e2.11c
e2.11d
e2.11e
Here, p′ designates the pressure deviation from the average over the off-wake region at the given vertical level.
To transform the above to a form closer to Eq. (31), we first approximate the kinetic energy by the vertical component within the convective-updraft subcomponent, that is,
e2.12a
and retain only the horizontal components at the boundary, namely
e2.12b
These approximations further suggest setting and . We also assume that the convective updraft boundary propagates with the gravity wave phase velocity—that is, VΓV* = C* (in the Doppler-shifted sense)—and we set C* · nu = C*. By furthermore setting p′ ≃ 0, εu ≃ 0, 〈wA ≃ 0, 〈w2A ≃ 0, we finally obtain
e2.13
where the two terms in the right-hand side are approximately given by
e2.14a
e2.14b
e2.15a
e2.15b
Equation (2.14a) is compared with their Eqs. (26) and (28), whereas Eq. (2.14b) is essentially identical to Eq. (29). Equation (2.13) even becomes closer to Eq. (31) when we assume MρσBwB, but it does not become identical.

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 Γj separating the jth 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.

For generality, we designate a general prognostic variable by φ and assume that it is governed by
eA.1
where 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/TCp.
A corresponding prognostic equation for the given variable φ for a jth subcomponent of subgrid-scale variability is obtained by integrating Eq. (A.1) over an area Σj occupied by the jth subcomponent:
eA.2
By incorporating the entrainment–detrainment hypothesis as well as the result (A.5) from the next section, the contour integral in Eq. (A.2) reduces to
eA.3
where j′ designates the subcomponent immediately surrounding the jth subcomponent. The result Eq. (A.3) can furthermore be substituted into Eq. (A.2), which gives
eA.4
Equations (2.5a) and (2.5b) are the special cases of Eq. (A.4).

b. Contributions from the gridbox boundary

Contributions from the gridbox boundary in the contour integral (A.3) are evaluated as follows. Note that each side of the gridbox boundary crosses the wake region for the length
eq1
where 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 = yL/2, x = xL/2, and y = y + L/2, designated by :
eq2
where . Consequently, we obtain
(A.5)

In the above derivation, we note that , , and σj 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 Vj, , and σj are assumed in the two middle expressions, indicating only the relevant coordinate in concern.

c. Mass budget equation

The mass budget equation is obtained by setting φ = 1 and F = 0 in Eq. (A.2):
eA.6
This leads to Eqs. (2.1a) and (2.1b). The contour integral in the above equation is obtained by setting φ = 1 in Eq. (A.3):
eA.7
By substitution of the above into Eq. (A.6), we obtain Eq. (2.4).

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.

A general form of the vertical velocity equation is given by
eA.8
Here, ρ and θ0 are reference vertical profiles for the density and the potential temperature, respectively. Here, θ′ designates a deviation of the potential temperature from the reference θ0. The subscript υ for indicating the virtual effect is omitted in the appendix in order to simplify the expressions.
By repeating the same reduction as for the general variable φ in section A of the appendix, we find an expression for the vertical velocity equation under SCA for the jth subcomponent:
eA.9
with 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).
An equation for the horizontal velocity V can also be introduced under SCA. The derivation is analogous to that of the vertical velocity, and it is given by
eA.10

e. Kinetic energy budget

The kinetic energy budget equation for the jth subcomponent is obtained by multiplying and with Eqs. (A.9) and (A.10), respectively, and by adding them together. The final result is obtained with the help of the mass budget equation (A.6).

Multiplication of with Eq. (A.9) leads to
eq3
We further add Eq. (A.6) multiplied by to the above. The calculation leads to
eq4
This is the kinetic energy budget for the vertical component.
The same is obtained for the horizontal component in an analogous manner. The result is
eq5
By taking a sum of vertical and horizontal components, we finally obtain
eA.11

REFERENCES

  • Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674701.

    • Search Google Scholar
    • Export Citation
  • Asai, T., and A. Kasahara, 1967: A theoretical study of the compensating downdraft motions associated with cumulus clouds. J. Atmos. Sci., 24, 487496.

    • Search Google Scholar
    • Export Citation
  • de Rooy, W. C., and A. P. Siebesma, 2010: Analytical expressions for entrainment and detrainment in cumulus convection. Quart. J. Roy. Meteor. Soc., 136, 12161227, doi:10.1002/qj.640.

    • Search Google Scholar
    • Export Citation
  • Grandpeix, J.-Y., and J.-P. Lafore, 2010: A density current parameterization coupled with Emanuel’s convection scheme. Part I: The models. J. Atmos. Sci., 67, 881897.

    • Search Google Scholar
    • Export Citation
  • Gyarmati, I., 1970: Non-Equilibrium Thermodynamics. Springer, 184 pp.

  • Majda, A. J., 2007: New multiscale models and self-similarity in tropical convection. J. Atmos. Sci., 64, 13931404.

  • Siebesma, A. P., 1998: Shallow cumulus convection. Buoyancy Convection in Geophysical Flows, E. J. Plate et al., Eds., Kluwer Academic, 441–486.

  • Yano, J.-I., 2009: Deep-convective vertical transport: What is mass flux? Atmos. Chem. Phys. Discuss., 9, 35353553.

  • Yano, J.-I., 2011: Interactive comment on “Simulating deep convection with a shallow convection scheme” by C. Hohenegger and C. S. Bretherton. Atmos. Chem. Phys. Discuss., 11, C2411–C2425.

  • Yano, J.-I., J.-L. Redelsperger, F. Guichard, and P. Bechtold, 2005: Mode decomposition as a methodology for developing convective-scale representations in global models. Quart. J. Roy. Meteor. Soc., 131, 23132336.

    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., P. Benard, F. Couvreux, and A. Lahellec, 2010: NAM-SCA: A nonhydrostatic anelastic model with segmentally constant approximations. Mon. Wea. Rev., 138, 19571974.

    • Search Google Scholar
    • Export Citation
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