• Bony, S., and K. A. Emanuel, 2001: A parameterization of the cloudiness associated with cumulus convection; Evaluation using TOGA COARE data. J. Atmos. Sci., 58 , 31583183.

    • Search Google Scholar
    • Export Citation
  • Bray, K. N. C., and P. A. Libby, 1994: Recent developments in the BML model of premixed turbulent combustion. Turbulent Reacting Flows, P. A. Libby and F. A. Williams, Eds., Academic Press, 115–151.

    • Search Google Scholar
    • Export Citation
  • Chen, J-Y., and W. Kollmann, 1994: Comparison of prediction and measurement in non-premixed turbulent flames. Turbulent Reacting Flows, P. A. Libby and F. A. Williams, Eds., Academic Press, 211–308.

    • Search Google Scholar
    • Export Citation
  • Colucci, P. J., F. A. Jaberi, P. Givi, and S. B. Pope, 1998: Filtered density function for large eddy simulation of turbulent reacting flows. Phys. Fluids, 10 , 499515.

    • Search Google Scholar
    • Export Citation
  • Cook, A. W., and J. J. Riley, 1994: A subgrid model for equilibrium chemistry in turbulent flows. Phys. Fluids, 6A , 28682870.

  • de Roode, S. R., P. G. Duynkerke, and A. P. Siebesma, 2000: Analogies between mass-flux and Reynolds-averaged equations. J. Atmos. Sci., 57 , 15851598.

    • Search Google Scholar
    • Export Citation
  • Dopazo, C., 1994: Recent developments in Pdf methods. Turbulent Reacting Flows, P. A. Libby and F. A. Williams, Eds., Academic Press, 375–474.

    • Search Google Scholar
    • Export Citation
  • Dopazo, C., and E. E. O'Brien, 1974: An approach to the autoignition of a turbulent mixture. Acta Astronaut., 1 , 12391266.

  • Fowler, L. D., D. A. Randall, and S. A. Rutledge, 1996: Liquid and ice cloud microphysics in the CSU general circulation model. Part I: Model description and simulated microphysical processes. J. Climate, 9 , 489529.

    • Search Google Scholar
    • Export Citation
  • Frankel, S. H., V. Adumitroaie, C. K. Madnia, and P. Givi, 1993: Large eddy simulation of turbulent reacting flow by assumed PDF methods. Engineering Applications of Large Eddy Simulations, S. A. Ragab and U. Piomelli, Eds., Vol. 162, ASME, 81–101.

    • Search Google Scholar
    • Export Citation
  • Genio, A. D. D., M. S. Yao, W. Kovari, and K. K. W. Lo, 1996: A prognostic cloud water parameterization for global climate models. J. Climate, 9 , 270304.

    • Search Google Scholar
    • Export Citation
  • Golaz, J-C., V. E. Larson, and W. R. Cotton, 2002a: A PDF-based model for boundary layer clouds. Part I: Method and model description. J. Atmos. Sci., 59 , 35403551.

    • Search Google Scholar
    • Export Citation
  • Golaz, J-C., V. E. Larson, and W. R. Cotton, 2002b: A PDF-based model for boundary layer clouds. Part II: Model results. J. Atmos. Sci., 59 , 35523571.

    • Search Google Scholar
    • Export Citation
  • Gregory, D., D. Wilson, and A. Bushell, 2002: Insights into cloud parameterization provided by a prognostic approach. Quart. J. Roy. Meteor. Soc., 128 , 14851504.

    • Search Google Scholar
    • Export Citation
  • Jakob, C., D. Gregory, and J. Teixeira, 1999: A package of cloud and convection changes for CY21R3. Res. Department Memo. R60.6.1/CJ/83, ECMWF, Reading, United Kingdom, 15 pp.

    • Search Google Scholar
    • Export Citation
  • Klimenko, A. Y., and R. W. Bilger, 1999: Conditional moment closure for turbulent combustion. Prog. Energy Combust. Sci., 25 , 595687.

    • Search Google Scholar
    • Export Citation
  • Lappen, C-L., and D. A. Randall, 2001a: Towards a unified parameterization of the boundary layer and moist convection. Part I: A new type of mass-flux model. J. Atmos. Sci., 58 , 20212036.

    • Search Google Scholar
    • Export Citation
  • Lappen, C-L., and D. A. Randall, 2001b: Towards a unified parameterization of the boundary layer and moist convection. Part II: Lateral mass exchanges and subplume-scale fluxes. J. Atmos. Sci., 58 , 20372051.

    • Search Google Scholar
    • Export Citation
  • Lappen, C-L., and D. A. Randall, 2001c: Towards a unified parameterization of the boundary layer and moist convection. Part III: Simulations of clear and cloudy convection. J. Atmos. Sci., 58 , 20522072.

    • Search Google Scholar
    • Export Citation
  • Larson, V. E., R. Wood, P. R. Field, J-C. Golaz, T. H. Vonder Haar, and W. R. Cotton, 2001a: Small-scale and mesoscale variability of scalars in cloudy boundary layers: One-dimensional probability density functions. J. Atmos. Sci., 58 , 19781994.

    • Search Google Scholar
    • Export Citation
  • Larson, V. E., R. Wood, P. R. Field, J-C. Golaz, T. H. Vonder Haar, and W. R. Cotton, 2001b: Systematic biases in the microphysics and thermodynamics of numerical models that ignore subgrid-scale variability. J. Atmos. Sci., 58 , 11171128.

    • Search Google Scholar
    • Export Citation
  • Larson, V. E., J-C. Golaz, and W. R. Cotton, 2002: Small-scale and mesoscale variability in cloudy boundary layers: Joint three-dimensional probability density functions. J. Atmos. Sci., 59 , 35193539.

    • Search Google Scholar
    • Export Citation
  • Lewellen, W. S., and S. Yoh, 1993: Binormal model of ensemble partial cloudiness. J. Atmos. Sci., 50 , 12281237.

  • Lundgren, T. S., 1967: Distribution functions in the statistical theory of turbulence. Phys. Fluids, 10 , 969975.

  • Lundgren, T. S., 1969: Model equation for nonhomogeneous turbulence. Phys. Fluids, 12 , 485497.

  • Manton, M. J., and W. R. Cotton, 1977: Formulation of approximate equations for modeling moist deep convection on the mesoscale. Atmospheric Science Paper No. 266 10, Colorado State University, Fort Collins, CO, 62 pp.

    • Search Google Scholar
    • Export Citation
  • Mellor, G. L., 1977: The Gaussian cloud model relations. J. Atmos. Sci., 34 , 356358.

  • Moeng, C-H., and D. A. Randall, 1984: Problems in simulating the stratocumulus-topped boundary layer with a third-order closure model. J. Atmos. Sci., 41 , 15881600.

    • Search Google Scholar
    • Export Citation
  • O'Brien, E. E., 1980: The probability density function (pdf) approach to reacting turbulent flows. Turbulent Reacting Flows, P. A. Libby and F. A. Williams, Eds., Springer-Verlag, 185–218.

    • Search Google Scholar
    • Export Citation
  • Pincus, R., and S. A. Klein, 2000: Unresolved spatial variability and microphysical process rates in large-scale models. J. Geophys. Res., 105 , 2705927065.

    • Search Google Scholar
    • Export Citation
  • Pope, S. B., 2000: Turbulent Flows. Cambridge University Press, 771 pp.

  • Pudykiewicz, J., R. Benoit, and J. Mailhot, 1992: Inclusion and verification of a predictive cloud-water scheme in a regional numerical weather prediction model. Mon. Wea. Rev., 120 , 612626.

    • Search Google Scholar
    • Export Citation
  • Randall, D. A., Q. Shao, and C-H. Moeng, 1992: A second-order bulk boundary-layer model. J. Atmos. Sci., 49 , 19031923.

  • Rotstayn, L. D., 2000: On the “tuning” of autoconversion parameterizations in climate models. J. Geophys. Res., 105 , 1549515507.

  • Sigg, R., 2000: Use of pseudoadiabatic adjustment of turbulence for a simplified nighttime stratocumulus case: A one-dimensional study. Mon. Wea. Rev., 128 , 23172328.

    • Search Google Scholar
    • Export Citation
  • Smith, R. N. B., 1990: A scheme for predicting layer clouds and their water content in a general circulation model. Quart. J. Roy. Meteor. Soc., 116 , 435460.

    • Search Google Scholar
    • Export Citation
  • Sommeria, G., and J. W. Deardorff, 1977: Subgrid-scale condensation in models of nonprecipitating clouds. J. Atmos. Sci., 34 , 344355.

    • Search Google Scholar
    • Export Citation
  • Stevens, B., W. R. Cotton, and G. Feingold, 1998: A critique of one- and two-dimensional models of boundary layer clouds with a binned representation of drop microphysics. Atmos. Res., 47 , –48. 529553.

    • Search Google Scholar
    • Export Citation
  • Sundqvist, H., 1978: A parameterization scheme for non-convective condensation including prediction of cloud water content. Quart. J. Roy. Meteor. Soc., 104 , 677690.

    • Search Google Scholar
    • Export Citation
  • Teixeira, J., 2001: Cloud fraction and relative humidity in a prognostic cloud fraction scheme. Mon. Wea. Rev., 129 , 17501753.

  • Tiedtke, M., 1993: Representation of clouds in large-scale models. Mon. Wea. Rev., 121 , 30403061.

  • Tompkins, A. M., 2002: A prognostic parameterization for the subgrid-scale variability of water vapor and clouds in large-scale models and its use to diagnose cloud cover. J. Atmos. Sci., 59 , 19171942.

    • Search Google Scholar
    • Export Citation
  • Villermaux, J., and J. C. Devillon, 1972: Représentation de la coalescence et de la redispersion des domaines de ségrégation dans un fluide per modèle d'interaction phénoménologique. Proc. Second Industrial Symp. on Chemical Reaction Engineering, Amsterdam, Netherlands.

    • Search Google Scholar
    • Export Citation
  • Wang, S., and Q. Wang, 1999: On condensation and evaporation in turbulence cloud parameterizations. J. Atmos. Sci., 56 , 33383344.

  • Wang, S., Q. Wang, and G. Feingold, 2003: Turbulence, condensation, and liquid water transport in numerically simulated nonprecipitating stratocumulus clouds. J. Atmos. Sci., 60 , 262278.

    • Search Google Scholar
    • Export Citation
  • Wilson, D., and D. Gregory, 2003: The behaviour of large-scale model cloud schemes under idealized forcing scenarios. Quart. J. Roy. Meteor. Soc., 129 , 967986.

    • Search Google Scholar
    • Export Citation
  • Zhao, Q., and F. H. Carr, 1997: A prognostic cloud scheme for operational NWP models. Mon. Wea. Rev., 125 , 19311953.

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Prognostic Equations for Cloud Fraction and Liquid Water, and Their Relation to Filtered Density Functions

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  • 1 Atmospheric Science Group, Department of Mathematical Sciences, University of Wisconsin—Milwaukee, Milwaukee, Wisconsin
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Abstract

This paper derives prognostic equations for cloud fraction and specific liquid water content. Such equations are carried by host models in order to provide more accurate forcing of microphysical and radiative parameterizations. The starting point of the derivations is a prognostic equation for the filtered density function (FDF) of a moisture variable. The FDF is the probability density function (PDF) of subgrid (i.e., filtered) fluctuations.

The resulting equations for cloud fraction and liquid water contain unclosed terms. Many of these terms can be closed if the relevant joint FDF is known. In addition, there emerge from the derivation two dissipation terms that are not closed even if the FDF of moisture is known. These dissipation terms do not appear in equations for conserved variables such as total water content.

The paper then compares various closures in the literature for turbulent flux, source, and dissipative mixing terms. Intuition suggests that dissipative mixing cannot usually increase liquid water content but that dissipative mixing can sometimes increase cloud fraction. The paper discusses two closures for dissipative mixing: one is based on the linear mean-square estimation (LMSE) model, and the other is due to Tiedtke. The LSME model permits cloud fraction to increase, as expected, but the Tiedtke model does not.

Corresponding author address: Vincent E. Larson, Department of Mathematical Sciences, University of Wisconsin—Milwaukee, P.O. Box 413, Milwaukee, WI 53201. Email: vlarson@uwm.edu

Abstract

This paper derives prognostic equations for cloud fraction and specific liquid water content. Such equations are carried by host models in order to provide more accurate forcing of microphysical and radiative parameterizations. The starting point of the derivations is a prognostic equation for the filtered density function (FDF) of a moisture variable. The FDF is the probability density function (PDF) of subgrid (i.e., filtered) fluctuations.

The resulting equations for cloud fraction and liquid water contain unclosed terms. Many of these terms can be closed if the relevant joint FDF is known. In addition, there emerge from the derivation two dissipation terms that are not closed even if the FDF of moisture is known. These dissipation terms do not appear in equations for conserved variables such as total water content.

The paper then compares various closures in the literature for turbulent flux, source, and dissipative mixing terms. Intuition suggests that dissipative mixing cannot usually increase liquid water content but that dissipative mixing can sometimes increase cloud fraction. The paper discusses two closures for dissipative mixing: one is based on the linear mean-square estimation (LMSE) model, and the other is due to Tiedtke. The LSME model permits cloud fraction to increase, as expected, but the Tiedtke model does not.

Corresponding author address: Vincent E. Larson, Department of Mathematical Sciences, University of Wisconsin—Milwaukee, P.O. Box 413, Milwaukee, WI 53201. Email: vlarson@uwm.edu

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