Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-10-31T23:00:09.185Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal models for precipitating turbulent convection

Published online by Cambridge University Press:  01 February 2013

Gerardo Hernandez-Duenas ,
Andrew J. Majda ,
Leslie M. Smith  and
Samuel N. Stechmann
Gerardo Hernandez-Duenas
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA
Andrew J. Majda
Affiliation:
Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1110, USA
Leslie M. Smith
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA Department of Engineering Physics, University of Wisconsin–Madison, Madison, WI 53706, USA
Samuel N. Stechmann*
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, WI 53706, USA
*
†Email address for correspondence: stechmann@wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Simulations of precipitating convection would typically use a non-Boussinesq dynamical core such as the anelastic equations, and would incorporate water substance in all of its phases: vapour, liquid and ice. Furthermore, the liquid water phase would be separated into cloud water (small droplets suspended in air) and rain water (larger droplets that fall). Depending on environmental conditions, the moist convection may organize itself on multiple length and time scales. Here we investigate the question, what is the minimal representation of water substance and dynamics that still reproduces the basic regimes of turbulent convective organization? The simplified models investigated here use a Boussinesq atmosphere with bulk cloud physics involving equations for water vapour and rain water only. As a first test of the minimal models, we investigate organization or lack thereof on relatively small length scales of approximately 100 km and time scales of a few days. It is demonstrated that the minimal models produce either unorganized (‘scattered’) or organized convection in appropriate environmental conditions, depending on the environmental wind shear. For the case of organized convection, the models qualitatively capture features of propagating squall lines that are observed in nature and in more comprehensive cloud resolving models, such as tilted rain water profiles, low-altitude cold pools and propagation speed corresponding to the maximum of the horizontally averaged, horizontal velocity.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Convection: Moist convection Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , pp. 576 - 611
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asai, T. 1970 Stability of a plane parallel flow with variable vertical shear and unstable stratification. J. Met. Soc. Japan 48, 129–139.Google Scholar
Bannon, P. R. 1996 On the anelastic approximation for a compressible atmosphere. J. Atmos. Sci. 53, 3618–3628.Google Scholar
Barnes, G. M. & Sieckman, K. 1984 The environment of fast-and slow-moving tropical mesoscale convective cloud lines. Mon. Weath. Rev. 112 (9), 1782–1794.Google Scholar
Bretherton, C. S. 1987 A theory for nonprecipitating moist convection between two parallel plates. Part I: thermodynamics and ‘linear’ solutions. J. Atmos. Sci. 44, 1809–1827.Google Scholar
Castaing, B., Gunaratne, G., Kadanoff, L., Libchaber, A. & Heslot, F. 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204 (1), 1–30.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Cuijpers, J. W. M. & Duynkerke, P. G. 1993 Large eddy simulation of trade wind cumulus clouds. J. Atmos. Sci. 50 (23), 3894–3908.Google Scholar
Deardorff, J. W. 1965 Gravitational instability between horizontal plates with shear. Phys. Fluids 8, 1027.Google Scholar
Deng, Q., Smith, L. M. & Majda, A. J. 2012 Tropical cyclogenesis and vertical shear in a moist Boussinesq model. J. Fluid Mech. 706, 384–412.Google Scholar
Emanuel, K. 1986 Some dynamical aspect of precipitating convection. J. Atmos. Sci. 43, 2183–2198.Google Scholar
Emanuel, K. A. 1994 Atmospheric Convection. Oxford University Press.CrossRefGoogle Scholar
Fovell, R. & Ogura, Y. 1988 Numerical simulation of a midlatitude squall line in two dimensions. J. Atmos. Sci. 45 (24), 3846–3879.Google Scholar
Grabowski, W. W. & Clark, T. L. 1993 Cloud–environment interface instability, part II: extension to three spatial dimensions. J. Atmos. Sci. 50, 555–573.Google Scholar
Grabowski, W. W. & Moncrieff, M. W. 2001 Large-scale organization of tropical convection in two-dimensional explicit numerical simulations. Q. J. R. Meteorol. Soc. 127, 445–468.Google Scholar
Grabowski, W. W. & Smolarkiewicz, P. K. 1996 Two-time-level semi-Lagrangian modeling of precipitating clouds. Mon. Weath. Rev. 124 (3), 487–497.Google Scholar
Grabowski, W. W., Wu, X. & Moncrieff, M. W. 1996 Cloud-resolving modeling of tropical cloud systems during Phase III of GATE. Part I: two-dimensional experiments. J. Atmos. Sci. 53, 3684–3709.Google Scholar
Grabowski, W. W., Wu, X., Moncrieff, M. W. & Hall, W. D. 1998 Cloud-resolving modeling of cloud systems during Phase III of GATE. Part II: effects of resolution and the third spatial dimension. J. Atmos. Sci. 55 (21), 3264–3282.Google Scholar
Hendon, H. H. & Liebmann, B. 1994 Organization of convection within the Madden–Julian oscillation. J. Geophys. Res. 99, 8073–8084.Google Scholar
Houze, R. A. 1993 Cloud Dynamics. Academic.Google Scholar
Houze, R. A. Jr. 2004 Mesoscale convective systems. Rev. Geophys. 42, G4003+.Google Scholar
Jorgensen, D. P., LeMone, M. A. & Trier, S. B. 1997 Structure and evolution of the 22 February 1993 TOGA COARE squall line: aircraft observations of precipitation, circulation, and surface energy fluxes. J. Atmos. Sci. 54 (15), 1961–1985.Google Scholar
Jung, J.-H. & Arakawa, A. 2005 Preliminary tests of multiscale modeling with a two-dimensional framework: sensitivity to coupling methods. Mon. Weath. Rev. 133 (3), 649–662.Google Scholar
Kessler, E. 1969 On the Distribution and Continuity of Water Substance in Atmospheric Circulations, Meteorological Monographs , vol. 32. American Meteorological Society.Google Scholar
Khouider, B., Han, Y., Majda, A. J. & Stechmann, S. N. 2012 Multiscale waves in an MJO background and convective momentum transport feedback. J. Atmos. Sci. 69, 915–933.Google Scholar
Kiladis, G. N., Wheeler, M. C., Haertel, P. T., Straub, K. H. & Roundy, P. E. 2009 Convectively coupled equatorial waves. Rev. Geophys. 47, RG2003.CrossRefGoogle Scholar
Kim, D., Sperber, K., Stern, W., Waliser, D., Kang, I.-S., Maloney, E., Wang, W., Weickmann, K., Benedict, J., Khairoutdinov, M., Lee, M.-I., Neale, R., Suarez, M., Thayer-Calder, K. & Zhang, G. 2009 Application of MJO simulation diagnostics to climate models. J. Climate 22 (23), 6413–6436.Google Scholar
Klein, R. & Majda, A. 2006 Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn. 20, 525–551.Google Scholar
Krishnamurti, R. 1970a On the transition to turbulent convection. Part 1. The transition from two-to three-dimensional flow. J. Fluid Mech. 42 (2), 295–307.Google Scholar
Krishnamurti, R. 1970b On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42 (2), 309–320.Google Scholar
Kuo, H. L. 1961 Convection in conditionally unstable atmosphere. Tellus 13 (4), 441–459.Google Scholar
Lafore, J. P. & Moncrieff, M. W. 1989 A numerical investigation of the organization and interaction of the convective and stratiform regions of tropical squall lines. J. Atmos. Sci. 46 (4), 521–544.Google Scholar
Lau, W. K. M. & Waliser, D. E. (Eds) 2011 Intraseasonal Variability in the Atmosphere–Ocean Climate System. Springer.Google Scholar
LeMone, M. A., Zipser, E. J. & Trier, S. B. 1998 The role of environmental shear and thermodynamic conditions in determining the structure and evolution of mesoscale convective systems during TOGA COARE. J. Atmos. Sci. 55 (23), 3493–3518.Google Scholar
Lilly, D. K. 1979 The dynamical structure and evolution of thunderstorms and squall lines. Annu. Rev. Earth Planet. Sci. 7, 117–161.Google Scholar
Lipps, F. B. & Hemler, R. S. 1982 A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 39, 2192–2210.2.0.CO;2>CrossRefGoogle Scholar
Liu, C. & Moncrieff, M. W. 2001 Cumulus ensembles in shear: implications for parameterization. J. Atmos. Sci. 58 (18), 2832–2842.Google Scholar
Lucas, C., Zipser, E. J. & Ferrier, B. S. 2000 Sensitivity of tropical west pacific oceanic squall lines to tropospheric wind and moisture profiles. J. Atmos. Sci. 57 (15), 2351–2373.Google Scholar
Majda, A. & Souganidis, P. 2000 The effect of turbulence on mixing in prototype reaction–diffusion systems. Commun. Pure Appl. Maths 53 (10), 1284–1304.Google Scholar
Majda, A. J. & Stechmann, S. N. 2008 Stochastic models for convective momentum transport. Proc. Natl Acad. Sci. U.S.A. 105, 17614–17619.Google Scholar
Majda, A. J. & Stechmann, S. N. 2009 A simple dynamical model with features of convective momentum transport. J. Atmos. Sci. 66, 373–392.Google Scholar
Majda, A. J. & Stechmann, S. N. 2011 Multiscale theories for the MJO. In Intraseasonal Variability in the Atmosphere–Ocean Climate System (ed. Lau, W. K. M. & Waliser, D. E.). Springer.Google Scholar
Majda, A. J. & Xing, Y. 2010 New multi-scale models on mesoscales and squall lines. Commun. Math. Sci. 8 (1), 113–144.CrossRefGoogle Scholar
Majda, A. J., Xing, Y. & Mohammadian, M. 2010 Moist multi-scale models for the hurricane embryo. J. Fluid Mech. 657, 478–501.Google Scholar
Mapes, B. E., Tulich, S., Lin, J.-L. & Zuidema, P. 2006 The mesoscale convection life cycle: building block or prototype for large-scale tropical waves?. Dyn. Atmos. Oceans 42, 3–29.Google Scholar
Moncrieff, M. W. 1981 A theory of organized steady convection and its transport properties. Q. J. R. Meteorol. Soc. 107 (451), 29–50.Google Scholar
Moncrieff, M. W. 1992 Organized convective systems: archetypal dynamical models, mass and momentum flux theory, and parameterization. Q. J. R. Meteorol. Soc. 118 (507), 819–850.Google Scholar
Moncrieff, M. W. 2010 The multiscale organization of moist convection and the intersection of weather and climate. In Climate Dynamics: Why Does Climate Vary? (ed. Sun, D.-Z. & Bryan, F.). Geophysical Monograph Series , vol. 189. pp. 3–26. American Geophysical Union.Google Scholar
Moncrieff, M. W. & Green, J. S. A. 1972 The propagation and transfer properties of steady convective overturning in shear. Q. J. R. Meteorol. Soc. 98 (416), 336–352.Google Scholar
Moncrieff, M. W. & Miller, M. J. 1976 The dynamics and simulation of tropical cumulonimbus and squall lines. Q. J. R. Meteorol. Soc. 102, 373–394.Google Scholar
Moncrieff, M. W., Shapiro, M., Slingo, J. & Molteni, F. 2007 Collaborative research at the intersection of weather and climate. WMO Bull. 56, 204–211.Google Scholar
Morrison, H. & Grabowski, W. W. 2008 Modeling supersaturation and subgrid-scale mixing with two-moment bulk warm microphysics. J. Atmos. Sci. 65 (3), 792–812.Google Scholar
Nakazawa, T. 1988 Tropical super clusters within intraseasonal variations over the Western Pacific. J. Met. Soc. Japan 66 (6), 823–839.Google Scholar
Ogura, Y. & Phillips, N. A. 1962 Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179.Google Scholar
Pauluis, O. 2008 Thermodynamic consistency of the anelastic approximation for a moist atmosphere. J. Atmos. Sci. 65 (8), 2719–2729.Google Scholar
Pauluis, O., Balaji, V. & Held, I. M. 2000 Frictional dissipation in a precipitating atmosphere. J. Atmos. Sci. 57 (7), 989–994.Google Scholar
Pauluis, O. & Dias, J. 2012 Satellite estimates of precipitation-induced dissipation in the atmosphere. Science 335 (6071), 953–956.Google Scholar
Pauluis, O. & Schumacher, J. 2010 Idealized moist Rayleigh–Bénard convection with piecewise linear equation of state. Commun. Math. Sci. 8, 295–319.Google Scholar
Pauluis, O. & Schumacher, J. 2011 Self-aggregation of clouds in conditionally unstable moist convection. Proc. Natl Acad. Sci. U.S.A. 108, 12623–12628.Google Scholar
Peters, N. 2000 Turbulent Combustion. Cambridge University Press.Google Scholar
Pruppacher, H. R. & Klett, J. D. 1997 Microphysics of Clouds and Precipitation. Kluwer Academic.Google Scholar
Rogers, R. R. & Yau, M. K. 1989 A Short Course in Cloud Physics. Butterworth–Heinemann.Google Scholar
Schumacher, J. & Pauluis, O. 2010 Buoyancy statistics in moist turbulent Rayleigh–Bénard convection. J. Fluid Mech. 648, 509–519.Google Scholar
Seitter, K. L. & Kuo, H.-L. 1983 The dynamical structure of squall-line type thunderstorms. J. Atmos. Sci. 40, 2831–2854.2.0.CO;2>CrossRefGoogle Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442–447.CrossRefGoogle Scholar
Spyksma, K. & Bartello, P. 2008 Small-scale moist turbulence in numerically generated convective clouds. J. Atmos. Sci. 65, 1967–1978.Google Scholar
Spyksma, K., Bartello, P. & Yau, M. K. 2006 A Boussinesq moist turbulence model. J. Turbul. 7, 1–24.Google Scholar
Stevens, B. 2005 Atmospheric moist convection. Annu. Rev. Earth Planet. Sci. 33 (1), 605–643.Google Scholar
Stevens, B. 2007 On the growth of layers of nonprecipitating cumulus convection. J. Atmos. Sci. 64, 2916–2931.Google Scholar
Straub, K. H., Haertel, P. T. & Kiladis, G. N. 2010 An analysis of convectively coupled Kelvin waves in 20 WCRP CMIP3 global coupled climate models. J. Climate 23 (11), 3031–3056.CrossRefGoogle Scholar
Sukhatme, J., Majda, A. J. & Smith, L. M. 2012 Two-dimensional moist stratified turbulence and the emergence of vertically sheared horizontal flows. Phys. Fluids 24, 036602.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press.Google Scholar
Wu, X., Grabowski, W. W. & Moncrieff, M. W. 1998 Long-term behavior of cloud systems in TOGA COARE and their interactions with radiative and surface processes. Part I: two-dimensional modeling study. J. Atmos. Sci. 55 (17), 2693–2714.Google Scholar
Wu, X. & Moncrieff, M. W. 1996 Collective effects of organized convection and their approximation in general circulation models. J. Atmos. Sci. 53 (10), 1477–1495.Google Scholar
Xu, K. M. & Randall, D. A. 1996 Explicit simulation of cumulus ensembles with the GATE Phase III data: comparison with observations. J. Atmos. Sci. 53, 3710–3736.Google Scholar
Zhang, C. 2005 Madden–Julian oscillation. Rev. Geophys. 43, G2003+.Google Scholar