• Arakawa, A., , and K.-M. Xu, 1992: The macroscopic behavior of simulated cumulus convection and semi-prognostic tests of the Arakawa–Schubert cumulus parameterization. Physical Processes in Atmospheric Models: Collection of Papers Presented at the Indo-U.S. Seminar on Parameterization of Subgrid-Scale Processes in Dynamical Models of Medium-Range Prediction and Global Climate, D. R. Sikka and S. Singh, Eds., John Wiley & Sons, 3–18.

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  • Lord, S. J., , and A. Arakawa, 1980: Interaction of a cumulus cloud ensemble with the large-scale environment. Part II. J. Atmos. Sci., 37, 26772692.

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  • Plant, S. R., , and J.-I. Yano, 2011: Comments on “An ensemble cumulus convection parameterization with explicit cloud treatment.” J. Atmos. Sci., 68, 15411544.

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  • Randall, D., , and D. Pan, 1993: Implementation of the Arakawa–Schubert cumulus parameterization with a prognostic closure. The Representation of Cumulus Convection in Numerical Models, Meteor. Monogr., No. 46, Amer. Meteor. Soc., 137–144.

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  • Wagner, T. M., , and H.-F. Graf, 2010: An ensemble cumulus convection parameterization with explicit cloud treatment. J. Atmos. Sci., 67, 38543869.

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  • 1 Atmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom
  • | 2 Centre for Atmospheric Science, University of Cambridge, Cambridge, United Kingdom
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Corresponding author address: Till M. Wagner, Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom. E-mail: till.wagner@physics.ox.ac.uk

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/2010JAS3485.1.

Corresponding author address: Till M. Wagner, Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom. E-mail: till.wagner@physics.ox.ac.uk

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/2010JAS3485.1.

In Wagner and Graf (2010, hereafter WG) we presented a convection parameterization based on the representation of 1) explicitly modeled individual cumulus clouds and 2) a cloud spectrum consisting of these different cumulus clouds. This approach was taken to address limitations of bulk mass flux parameterizations such as the incorporation of aerosol effects in convective clouds, mixed phase microphysics, and cloud coverage by convective clouds of different heights. The presented parameterization delivers realistic spectra of vertical velocities, precipitation intensities, entrainment and detrainment profiles, and cloud coverage, when compared to atmospheric measurements.

Plant and Yano (2011, hereafter PY) present an alternative derivation of the closure formulation on which we will comment in the following.

In section 3a PY claim that the damping time scale τA of the cloud work function has to be much larger than the time scale describing the dissipation of the convective kinetic energy (i.e., τAτdis). This statement is based on the inspection of the perturbation equations corresponding to the system consisting of PY’s Eqs. (2) and (5), leading to their Eqs. (A1) and (A2), and the perturbation equation corresponding to PY’s Eq. (14), leading to their Eq. (A6):
e1
The system of Eqs. (A1) and (A2) is solved to yield PY’s Eq. (A4):
e2
where τA = α/(dis), describing the decay rate η of the system. PY state that η as given by Eq. (2) agrees with η as given by Eq. (1) only if τAτdis. We shall discuss this argument as follows.
Under the assumption of τAτdis, Eq. (2) reduces to
eq1
because τdis/τA vanishes. Equation (1), however, leads immediately to the damping rate of η = 1/τA, which clearly contradicts the statement that both damping rates agree if τAτdis. In other words, if the damping scale of the cloud work function is defined as τA = α/(dis) [cf. PY’s Eq. (A5)] and the exponential decay rates η of Eqs. (1) and (2) [PY’s (A4) and (A6)] should agree, we find that τAτdis.

Therefore, we cannot follow the argument of PY that the necessity of the time scale separation poses a limiting factor of our model. Moreover, we do not regard convective kinetic energy quasi-equilibrium and cloud work function quasi-equilibrium as two conceptually different notions, since both hypotheses state that the large-scale forcing and the cumulus cloud population vary in time in a coupled way. This fact is elucidated by the explicit derivation of the cloud work function quasi-equilibrium from the assumption of the kinetic energy quasi-equilibrium by Lord and Arakawa (1980). Therefore, we presume that the forward integration of the population dynamics equation under the assumption of kinetic energy quasi-equilibrium or “convective equilibrium of the second kind,” as termed by PY, will result in a cloud population distribution in or close to an equilibrium amount.

To arrive at this equilibrium amount of clouds we construct a virtual time scale, where we hold the individual clouds as well as the environment constant and only vary the cloud numbers. This is a numerical technique called “virtual time-stepping” that is widely used in engineering applications. The resulting cloud distribution is taken as the cloud spectrum for this time step. In this virtual time scale, where solely the cloud numbers change and the cloud work function, which only depends on the individual cloud and the environment, does not change, we have dA/dtdM/dt. On the time scale of the evolution of the host model, however, where the environment as well as the individual cloud characteristics, and thus the cloud distribution, is updated at every large-scale time step, the cloud work function and the cloud distribution evolve together.

We appreciate the constructive comment of PY (cf. section 3c of PY) that the relationship between convective kinetic energy and cloud-base mass flux suggested by Arakawa and Xu (1992) and Randall and Pan (1993), with αi = const—assumes that the cloud cover fraction σi is constant. Thus, under the assumption of clouds in steady state αi should not be constant but rather should depend on the cloud cover fraction σi. PY suggest employing the more general relationship between convective kinetic energy and cloud-base mass flux , with ci = const and 1 < p < 2. In this case, however, instead of being constant, as argued by PY, ci would depend not only on cloud fraction σi but additionally on the vertical velocity wi, which would add further terms to the closure formulation. In WG we followed the simplest approach and assumed αi to be constant.

The aim of WG is to find a computationally efficient estimate of the individual cloud amounts under the convective equilibrium assumption. Experiments using the approach of inverting the interaction matrix proved to be computationally much more expensive than forward integration because of the necessity of performing multiple matrix inversions in most cases given that negative cloud numbers are physically not reasonable. In these tests we removed the equation associated with the cloud type holding a negative amount and moved to the reduced system. Since the forward integration of the dynamical system produced realistic results we chose this closure formulation.

Acknowledgments

This study has benefited from discussions through the COST Action ES0905.

REFERENCES

  • Arakawa, A., , and K.-M. Xu, 1992: The macroscopic behavior of simulated cumulus convection and semi-prognostic tests of the Arakawa–Schubert cumulus parameterization. Physical Processes in Atmospheric Models: Collection of Papers Presented at the Indo-U.S. Seminar on Parameterization of Subgrid-Scale Processes in Dynamical Models of Medium-Range Prediction and Global Climate, D. R. Sikka and S. Singh, Eds., John Wiley & Sons, 3–18.

    • Search Google Scholar
    • Export Citation
  • Lord, S. J., , and A. Arakawa, 1980: Interaction of a cumulus cloud ensemble with the large-scale environment. Part II. J. Atmos. Sci., 37, 26772692.

    • Search Google Scholar
    • Export Citation
  • Plant, S. R., , and J.-I. Yano, 2011: Comments on “An ensemble cumulus convection parameterization with explicit cloud treatment.” J. Atmos. Sci., 68, 15411544.

    • Search Google Scholar
    • Export Citation
  • Randall, D., , and D. Pan, 1993: Implementation of the Arakawa–Schubert cumulus parameterization with a prognostic closure. The Representation of Cumulus Convection in Numerical Models, Meteor. Monogr., No. 46, Amer. Meteor. Soc., 137–144.

    • Search Google Scholar
    • Export Citation
  • Wagner, T. M., , and H.-F. Graf, 2010: An ensemble cumulus convection parameterization with explicit cloud treatment. J. Atmos. Sci., 67, 38543869.

    • Search Google Scholar
    • Export Citation
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