Editorial Type:
Article
Article Type:
Research Article

Theoretical Expressions for the Ascent Rate of Moist Deep Convective Thermals

Hugh Morrison National Center for Atmospheric Research, Boulder, Colorado

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John M. Peters Department of Meteorology, U.S. Naval Postgraduate School, Monterey, California

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Print Publication:
01 May 2018
DOI:
https://doi.org/10.1175/JAS-D-17-0295.1
Page(s):
1699–1719
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Abstract

An approximate analytic expression is derived for the ratio λ of the ascent rate of moist deep convective thermals and the maximum vertical velocity within them; λ is characterized as a function of two nondimensional buoyancy-dependent parameters y and h and is used to express the thermal ascent rate as a function of the buoyancy field. The parameter y characterizes the vertical distribution of buoyancy within the thermal, and h is the ratio of the vertically integrated buoyancy from the surface to the thermal top and the vertical integral of buoyancy within the thermal. Theoretical λ values are calculated using values of y and h obtained from idealized numerical simulations of ascending moist updrafts and compared to λ computed directly from the simulations. The theoretical values of 0.4–0.8 are in reasonable agreement with the simulated λ (correlation coefficient of 0.86). These values are notably larger than the from Hill’s (nonbuoyant) analytic spherical vortex, which has been used previously as a framework for understanding the dynamics of moist convective thermals. The relatively large values of λ are a result of net positive buoyancy within the upper part of thermals that opposes the downward-directed dynamic pressure gradient force below the thermal top. These results suggest that nonzero buoyancy within moist convective thermals, relative to their environment, fundamentally alters the relationship between the maximum vertical velocity and the thermal-top ascent rate compared to nonbuoyant vortices. Implications for convection parameterizations and interpretation of the forces contributing to thermal drag are discussed.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: H. Morrison, morrison@ucar.edu

Abstract

An approximate analytic expression is derived for the ratio λ of the ascent rate of moist deep convective thermals and the maximum vertical velocity within them; λ is characterized as a function of two nondimensional buoyancy-dependent parameters y and h and is used to express the thermal ascent rate as a function of the buoyancy field. The parameter y characterizes the vertical distribution of buoyancy within the thermal, and h is the ratio of the vertically integrated buoyancy from the surface to the thermal top and the vertical integral of buoyancy within the thermal. Theoretical λ values are calculated using values of y and h obtained from idealized numerical simulations of ascending moist updrafts and compared to λ computed directly from the simulations. The theoretical values of 0.4–0.8 are in reasonable agreement with the simulated λ (correlation coefficient of 0.86). These values are notably larger than the from Hill’s (nonbuoyant) analytic spherical vortex, which has been used previously as a framework for understanding the dynamics of moist convective thermals. The relatively large values of λ are a result of net positive buoyancy within the upper part of thermals that opposes the downward-directed dynamic pressure gradient force below the thermal top. These results suggest that nonzero buoyancy within moist convective thermals, relative to their environment, fundamentally alters the relationship between the maximum vertical velocity and the thermal-top ascent rate compared to nonbuoyant vortices. Implications for convection parameterizations and interpretation of the forces contributing to thermal drag are discussed.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: H. Morrison, morrison@ucar.edu
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  • Austin, P. H., M. B. Baker, A. M. Blyth, and J. B. Jensen, 1985: Small-scale variability in warm continental cumulus clouds. J. Atmos. Sci., 42, 1123–1138, https://doi.org/10.1175/1520-0469(1985)042<1123:SSVIWC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Blyth, A. M., and J. Latham, 1993: Development of ice and precipitation in New Mexican summertime cumulus clouds. Quart. J. Roy. Meteor. Soc., 119, 91–120, https://doi.org/10.1002/qj.49711950905.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Blyth, A. M., W. A. Cooper, and J. B. Jensen, 1988: A study of the source of entrained air in Montana cumuli. J. Atmos. Sci., 45, 3944–3964, https://doi.org/10.1175/1520-0469(1988)045<3944:ASOTSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist updrafts. Mon. Wea. Rev., 131, 2394–2416, https://doi.org/10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chow, F. K., R. L. Street, M. Xue, and J. H. Feriger, 2005: Explicit filtering and reconstruction turbulence modeling for large-eddy simulation of neutral boundary layer flow. J. Atmos. Sci., 62, 2058–2077, https://doi.org/10.1175/JAS3456.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Damiani, R., G. Vali, and S. Haimov, 2006: The structure of thermals in cumulus from airborne dual-Doppler radar observations. J. Atmos. Sci., 63, 1432–1450, https://doi.org/10.1175/JAS3701.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Davies-Jones, R., 2003: An expression for effective buoyancy in surroundings with horizontal density gradients. J. Atmos. Sci., 60, 2922–2925, https://doi.org/10.1175/1520-0469(2003)060<2922:AEFEBI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2011: The influence of the cloud shell on tracer budget measurements of LES cloud entrainment. J. Atmos. Sci., 68, 2909–2920, https://doi.org/10.1175/2011JAS3658.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • de Roode, S. R., A. P. Siebesma, H. J. J. Jonker, and Y. de Voogd, 2012: Parameterization of the vertical velocity equation for shallow convection. Mon. Wea. Rev., 140, 2424–2436, https://doi.org/10.1175/MWR-D-11-00277.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • de Rooy, W. C., and Coauthors, 2013: Entrainment and detrainment in cumulus convection: An overview. Quart. J. Roy. Meteor. Soc., 139, 1–19, https://doi.org/10.1002/qj.1959.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hernandez-Deckers, D., and S. C. Sherwood, 2016: A numerical investigation of cumulus thermals. J. Atmos. Sci., 73, 4117–4136, https://doi.org/10.1175/JAS-D-15-0385.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Heus, T., C. F. J. Pols, H. J. J. Jonker, H. E. A. van den Akker, E. J. Griffith, M. Koutek, and F. H. Post, 2009: A statistical approach to the life cycle analysis of cumulus clouds selected in a virtual reality environment. J. Geophys. Res., 114, D06208, https://doi.org/10.1029/2008JD010917.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hill, M. J. M., 1894: On a spherical vortex. Philos. Trans. Roy. Soc. London, 185A, 213–245, https://doi.org/10.1098/rsta.1894.0006.

  • Jeevanjee, N., 2017: Vertical velocity in the gray zone. J. Adv. Model. Earth Sci., 9, 2304–2316, https://doi.org/10.1002/2017MS001059.

  • Jeevanjee, N., and D. M. Romps, 2016: Effective buoyancy at the surface and aloft. Quart. J. Roy. Meteor. Soc., 142, 811–820, https://doi.org/10.1002/qj.2683.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jonker, H. J. J., T. Heus, and P. P. Sullivan, 2008: A refined view of vertical mass transport by cumulus convection. J. Geophys. Res., 35, L07810, https://doi.org/10.1029/2007GL032606.

    • Search Google Scholar
    • Export Citation

  • Lamb, H., 1932: Hydrodynamics. Cambridge University Press, 738 pp.

  • Lasher-Trapp, S., D. C. Leon, P. J. DeMott, C. M. Villanuva-Birriel, A. V. Johnson, D. H. Moser, C. S. Tully, and W. Wu, 2016: A multisensor investigation of rime splintering in tropical maritime cumuli. J. Atmos. Sci., 73, 2547–2564, https://doi.org/10.1175/JAS-D-15-0285.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lebo, Z. J., and H. Morrison, 2015: Effects of horizontal and vertical grid spacing on mixing in simulated squall lines and implications for convective strength and structure. Mon. Wea. Rev., 143, 4355–4375, https://doi.org/10.1175/MWR-D-15-0154.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Levine, J., 1959: Spherical vortex theory of bubble-like motion in cumulus clouds. J. Meteor., 16, 653–662, https://doi.org/10.1175/1520-0469(1959)016<0653:SVTOBL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ludlam, F. H., and R. S. Scorer, 1953: Reviews of modern meteorology—10: Convection in the atmosphere. Quart. J. Roy. Meteor. Soc., 79, 317–341, https://doi.org/10.1002/qj.49707934102.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Malkus, J. S., and R. S. Scorer, 1955: The erosion of cumulus towers. J. Meteor., 12, 43–57, https://doi.org/10.1175/1520-0469(1955)012<0000:TEOCT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Morrison, H., 2016a: Impacts of updraft size and dimensionality on the perturbation pressure and vertical velocity in cumulus convection. Part I: Simple, generalized analytic solutions. J. Atmos. Sci., 73, 1441–1454, https://doi.org/10.1175/JAS-D-15-0040.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Morrison, H., 2016b: Impacts of updraft size and dimensionality on the perturbation pressure and vertical velocity in cumulus convection. Part II: Comparison of theoretical and numerical solutions. J. Atmos. Sci., 73, 1455–1480, https://doi.org/10.1175/JAS-D-15-0041.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Morrison, H., 2017: An analytic description of the structure and evolution of growing deep cumulus updrafts. J. Atmos. Sci., 74, 809–834, https://doi.org/10.1175/JAS-D-16-0234.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62, 1665–1677, https://doi.org/10.1175/JAS3446.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Morton, B. R., 1957: Buoyant plumes in a moist atmosphere. J. Fluid Mech., 2, 127–144, https://doi.org/10.1017/S0022112057000038.

  • Morton, B. R., G. Taylor, and J. S. Turner, 1956: Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. London, 234A, 1–23, https://doi.org/10.1098/rspa.1956.0011.

    • Search Google Scholar
    • Export Citation

  • Moser, D. H., and S. Lasher-Trapp, 2017: The influence of successive thermals on entrainment and dilution in a simulated cumulus congestus. J. Atmos. Sci., 74, 375–392, https://doi.org/10.1175/JAS-D-16-0144.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Neggers, R. A. J., A. P. Siebesma, and H. J. J. Jonker, 2002: A multiparcel method for shallow cumulus convection. J. Atmos. Sci., 59, 1655–1668, https://doi.org/10.1175/1520-0469(2002)059<1655:AMMFSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Peters, J. M., 2016: The impact of effective buoyancy and dynamic pressure forcing on vertical velocities within two-dimensional updrafts. J. Atmos. Sci., 73, 4531–4551, https://doi.org/10.1175/JAS-D-16-0016.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Romps, D. M., and A. B. Charn, 2015: Sticky thermals: Evidence for a dominant balance between buoyancy and drag in cloud updrafts. J. Atmos. Sci., 72, 2890–2901, https://doi.org/10.1175/JAS-D-15-0042.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Romps, D. M., and R. Oktem, 2015: Stereo photogrammetry reveals substantial drag on cloud thermals. Geophys. Res. Lett., 42, 5051–5057, https://doi.org/10.1002/2015GL064009.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sanchez, O., D. J. Raymond, L. Libersky, and A. G. Petschek, 1989: The development of thermals from rest. J. Atmos. Sci., 46, 2280–2292, https://doi.org/10.1175/1520-0469(1989)046<2280:TDOTFR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Scorer, R. S., 1957: Experiments on convection of isolated masses of buoyant fluid. J. Fluid Mech., 2, 583–594, https://doi.org/10.1017/S0022112057000397.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Scorer, R. S., and F. H. Ludlam, 1953: Bubble theory of penetrative convection. Quart. J. Roy. Meteor. Soc., 79, 94–103, https://doi.org/10.1002/qj.49707933908.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection. J. Atmos. Sci., 52, 650–656, https://doi.org/10.1175/1520-0469(1995)052<0650:EOPAFS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and H. J. J. Jonker, 2000: Anomalous scaling of cumulus cloud boundaries. Phys. Rev. Lett., 85, 214–217, https://doi.org/10.1103/PhysRevLett.85.214.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sherwood, S. C., D. Hernandez-Deckers, and M. Colin, 2013: Slippery thermals and the cumulus entrainment paradox. J. Atmos. Sci., 70, 2426–2442, https://doi.org/10.1175/JAS-D-12-0220.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Simpson, J., and V. Wiggert, 1969: Models of precipitating cumulus towers. Mon. Wea. Rev., 97, 471–489, https://doi.org/10.1175/1520-0493(1969)097<0471:MOPCT>2.3.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Smagorinsky, J., 1963: General circulation experiments with the primitive equations. Mon. Wea. Rev., 91, 99–164, https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Stevens, B., and Coauthors, 2001: Simulations of trade wind cumuli under a strong inversion. J. Atmos. Sci., 58, 1870–1891, https://doi.org/10.1175/1520-0469(2001)058<1870:SOTWCU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turner, J. S., 1962: The “starting plume” in neutral surroundings. J. Fluid Mech., 13, 356–368, https://doi.org/10.1017/S0022112062000762.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turner, J. S., 1963: Model experiments relating to thermals with increasing buoyancy. Quart. J. Roy. Meteor. Soc., 89, 62–74, https://doi.org/10.1002/qj.49708937904.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turner, J. S., 1964: The flow into an expanding thermal. J. Fluid Mech., 18, 195–208, https://doi.org/10.1017/S0022112064000155.

  • Warren, F. W. G., 1960: Wave resistance to vertical motion in a stratified fluid. J. Fluid Mech., 7, 209–229, https://doi.org/10.1017/S0022112060001444.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Weisman, M. L., and J. B. Klemp, 1982: The dependence of numerically simulated convective storms on vertical wind shear and buoyancy. Mon. Wea. Rev., 110, 504–520, https://doi.org/10.1175/1520-0493(1982)110<0504:TDONSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Woodward, B., 1959: The motion in and around isolated thermals. Quart. J. Roy. Meteor. Soc., 85, 144–151, https://doi.org/10.1002/qj.49708536407.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yano, J.-I., 2014: Basic convective element: Bubble or plume? A historical review. Atmos. Chem. Phys., 14, 7019–7030, https://doi.org/10.5194/acp-14-7019-2014.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, M., and P. H. Austin, 2005: Life cycle of numerically simulated shallow cumulus clouds. Part I: Transport. J. Atmos. Sci., 62, 1269–1290, https://doi.org/10.1175/JAS3414.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
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