The Behavior of Number Concentration Tendencies for the Continuous Collection Growth Equation Using One- and Two-Moment Bulk Parameterization Schemes

Jerry M. Straka School of Meteorology, University of Oklahoma, Norman, Oklahoma

Search for other papers by Jerry M. Straka in
Current site
Google Scholar
PubMed
Close
,
Katharine M. Kanak Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

Search for other papers by Katharine M. Kanak in
Current site
Google Scholar
PubMed
Close
, and
Matthew S. Gilmore Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois

Search for other papers by Matthew S. Gilmore in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

This paper presents a mathematical explanation for the nonconservation of total number concentration Nt of hydrometeors for the continuous collection growth process, for which Nt physically should be conserved for selected one- and two-moment bulk parameterization schemes. Where possible, physical explanations are proposed. The assumption of a constant no in scheme A is physically inconsistent with the continuous collection growth process, as is the assumption of a constant Dn for scheme B. Scheme E also is nonconservative, but it seems this result is not because of a physically inconsistent specification; rather the solution scheme’s equations simply do not satisfy Nt conservation and Nt does not come into the derivation. Even scheme F, which perfectly conserves Nt, does not preserve the distribution shape in comparison with a bin model.

Corresponding author address: Jerry M. Straka, University of Oklahoma, School of Meteorology, 120 David L. Boren Blvd., Norman, OK 73072. Email: jmstraka@cox.net

Abstract

This paper presents a mathematical explanation for the nonconservation of total number concentration Nt of hydrometeors for the continuous collection growth process, for which Nt physically should be conserved for selected one- and two-moment bulk parameterization schemes. Where possible, physical explanations are proposed. The assumption of a constant no in scheme A is physically inconsistent with the continuous collection growth process, as is the assumption of a constant Dn for scheme B. Scheme E also is nonconservative, but it seems this result is not because of a physically inconsistent specification; rather the solution scheme’s equations simply do not satisfy Nt conservation and Nt does not come into the derivation. Even scheme F, which perfectly conserves Nt, does not preserve the distribution shape in comparison with a bin model.

Corresponding author address: Jerry M. Straka, University of Oklahoma, School of Meteorology, 120 David L. Boren Blvd., Norman, OK 73072. Email: jmstraka@cox.net

Save
  • Carrio, G. G., and M. Nicolini, 1999: A double moment warm rain scheme: Description and test within a kinematic framework. Atmos. Res., 52 , 116.

    • Search Google Scholar
    • Export Citation
  • Cohard, J-M., and J-P. Pinty, 2000: A comprehensive two-moment warm microphysical bulk scheme. I: Description and tests. Quart. J. Roy. Meteor. Soc., 126 , 18151842.

    • Search Google Scholar
    • Export Citation
  • Ferrier, B. S., 1994: A double-moment multiple-phase four-class bulk ice scheme. Part I: Description. J. Atmos. Sci., 51 , 249280.

  • Gilmore, M. S., J. M. Straka, and E. N. Rasmussen, 2004a: Precipitation and evolution sensitivity in simulated deep convective storms: Comparisons between liquid-only and simple ice and liquid phase microphysics. Mon. Wea. Rev., 132 , 18971916.

    • Search Google Scholar
    • Export Citation
  • Gilmore, M. S., J. M. Straka, and E. N. Rasmussen, 2004b: Precipitation uncertainty due to variations in precipitation particle parameters within a simple microphysics scheme. Mon. Wea. Rev., 132 , 26102627.

    • Search Google Scholar
    • Export Citation
  • Koenig, L. R., and F. W. Murray, 1976: Ice-bearing cumulus cloud evolution: Numerical simulation and general comparison against observations. J. Appl. Meteor., 15 , 747762.

    • Search Google Scholar
    • Export Citation
  • Lin, Y-L., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Climate Appl. Meteor., 22 , 10651092.

    • Search Google Scholar
    • Export Citation
  • Meyers, P. M., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud physics parameterization. Part II: The two-moment scheme. Atmos. Res., 45 , 339.

    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., 1994: A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part I: Microphysics. J. Atmos. Sci., 51 , 797816.

    • Search Google Scholar
    • Export Citation
  • Passarelli, R. E., 1978: An approximate analytical model of the vapor deposition and aggregation growth of snowflakes. J. Atmos. Sci., 35 , 118124.

    • Search Google Scholar
    • Export Citation
  • Passarelli, R. E., and R. C. Srivastava, 1979: A new aspect of snowflake aggregation theory. J. Atmos. Sci., 36 , 484493.

  • Reisner, J., R. M. Rasmussen, and R. T. Bruintjes, 1998: Explicit forecasting of supercooled liquid water in winter storms using the MM5 model. Quart. J. Roy. Meteor. Soc., 124 , 10711107.

    • Search Google Scholar
    • Export Citation
  • Rutledge, S. A., and P. V. Hobbs, 1983: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. VIII: A model for the “seeder-feeder” process in warm-frontal rainbands. J. Atmos. Sci., 40 , 11851206.

    • Search Google Scholar
    • Export Citation
  • Srivastava, R. C., 1978: Parameterization of raindrop size distributions. J. Atmos. Sci., 35 , 108117.

  • Stalker, J. R., and K. R. Knupp, 2003: Cell merger potential in multicell thunderstorms of weakly sheared environments: Cell separation distance versus planetary boundary layer depth. Mon. Wea. Rev., 131 , 16781695.

    • Search Google Scholar
    • Export Citation
  • Straka, J. M., M. S. Gilmore, K. M. Kanak, and E. N. Rasmussen, 2005: A comparison of the conservation of number concentration for the continuous collection and vapor diffusion growth equations using one- and two-moment schemes. J. Appl. Meteor., 44 , 18441849.

    • Search Google Scholar
    • Export Citation
  • van den Heever, S. C., and W. R. Cotton, 2004: The impact of hail size on simulated supercell storms. J. Atmos. Sci., 61 , 15961609.

  • Verlinde, J., and W. R. Cotton, 1993: Fitting microphysical observations of nonsteady convective clouds to a numerical model: An application of the adjoint technique of data assimilation to a kinematic model. Mon. Wea. Rev., 121 , 27762793.

    • Search Google Scholar
    • Export Citation
  • Walko, R. L., W. R. Cotton, M. P. Meyers, and J. Y. Harrington, 1995: New RAMS cloud physics parameterization. Part I: The single-moment scheme. Atmos. Res., 38 , 339.

    • Search Google Scholar
    • Export Citation
  • Ziegler, C. L., 1985: Retrieval of thermal and microphysical variables in observed convective storms. Part 1: Model development and preliminary testing. J. Atmos. Sci., 42 , 14871509.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 133 88 0
PDF Downloads 26 12 0