Study of low-order numerical effects in the two-dimensional cloud-top mixing layer


Large-eddy simulation (LES) has been extensively used as a tool to understand how various processes contribute to the dynamics of the stratocumulus layer. These studies are complicated by the fact that many processes are tied to the dynamics of the stably stratified interface that caps the stratocumulus layer, and which is inadequately resolved by LES. Recent direct numerical simulations (DNS) of isobaric mixing due to buoyancy reversal in a cloud-top mixing layer show that molecular effects are in some instances important in setting the cloud-top entrainment rate, which in turn influences the global development of the layer. This suggests that traditional LES are fundamentally incapable of representing cloud-top processes that depend on buoyancy reversal and that numerical artefacts can affect significantly the results. In this study, we investigate a central aspect of this issue by developing a test case that embodies important features of the buoyancy-reversing cloud-top layer. So doing facilitates a one-to-one comparison of the numerical algorithms typical of LES and DNS codes in a well-established case. We focus on the numerical effects only by switching off the subgrid-scale model in the LES code and using instead a molecular viscosity. We systematically refine the numerical grid and quantify numerical errors, validate convergence and assess computational efficiency of the low-order LES code compared to the high-order DNS. We show that the high-order scheme solves the cloud-top problem computationally more efficiently. On that basis, we suggest that the use of higher-order schemes might be more attractive than further increasing resolution to improve the representation of stratocumulus in LES.


  1. 1

    Bony S.: Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models. Geophys. Res. Lett. 32(20), L20806 (2005)

    Article  Google Scholar 

  2. 2

    Deardorff J.W.: Cloud top entrainment instability. J. Atmos. Sci. 37, 131–147 (1980)

    Article  Google Scholar 

  3. 3

    Krueger S.: Linear eddy modeling of entrainment and mixing in stratus clouds. J. Atmos. Sci. 50(18), 3078–3090 (1993)

    Article  Google Scholar 

  4. 4

    Lele S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5

    Lilly D.K.: Models of cloud-topped mixed layers under a strong inversion. Q. J. R. Meteorol. Soc. 94(401), 292–309 (1968)

    Article  Google Scholar 

  6. 6

    Lomax H., Pulliam T.H., Zingg D.W.: Fundamentals of Computational Fluid Dynamics. Springer, Berlin (1998)

    Google Scholar 

  7. 7

    Mellado J.P.: The evaporatively driven cloud-top mixing layer. J. Fluid Mech. 660, 5–36 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8

    Mellado J.P., Stevens B., Schmidt H., Peters N.: Buoyancy reversal in cloud-top mixing layers. Q. J. R. Meteorol. Soc. 135, 963–978 (2009)

    Article  Google Scholar 

  9. 9

    Mellado J.P., Stevens B., Schmidt H., Peters N.: Probability density functions in the cloud-top mixing layer. New J. Phys. 12(8), 085010 (2010)

    Article  Google Scholar 

  10. 10

    Mellado J.P., Stevens B., Schmidt H., Peters N.: Two-fluid formulation of the cloud-top mixing layer. Theor. Comput. Fluid Dyn. 24(6), 511–536 (2010)

    MATH  Article  Google Scholar 

  11. 11

    Moeng C.H., Stevens B., Sullivan P.P.: Where is the interface of the stratocumulus-topped PBL?. J. Atmos. Sci. 62(7), 2626–2631 (2005)

    Article  Google Scholar 

  12. 12

    Ogura Y., Phillips N.A.: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19(2), 173–179 (1962)

    Article  Google Scholar 

  13. 13

    Pope S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)

    MATH  Book  Google Scholar 

  14. 14

    Randall D.A.: Conditional instability of the first kind upside-down. J. Atmos. Sci. 37, 125–130 (1980)

    Article  Google Scholar 

  15. 15

    Shy S.S., Breidenthal R.E.: Laboratory experiments on the cloud-top entrainment instability. J. Fluid Mech. 214(1), 1–15 (1990)

    Article  Google Scholar 

  16. 16

    Siems S.T., Bretherton C.S., Baker M.B., Shy S.S., Breidenthal R.E.: Buoyancy reversal and cloud-top entrainment instability. Q. J. R. Meteorol. Soc. 116(493), 705–739 (1990)

    Article  Google Scholar 

  17. 17

    Stevens B.: Entrainment in stratocumulus-topped mixed layers. Q. J. R. Meteorol. Soc. 128(586), 2663–2690 (2002)

    Article  Google Scholar 

  18. 18

    Stevens B., Lenschow D.H., Vali G., Gerber H., Bandy A., Blomquist B., Brenguier J.L., Bretherton C.S., Burnet F., Campos T., Chai S., Faloona I., Friesen D., Haimov S., Laursen K., Lilly D.K., Loehrer S.M., Malinowski S.P., Morley B., Petters M.D., Rogers D.C., Russell L., Savic-Jovcic V., Snider J.R., Straub D., Szumowski M.J., Takagi H., Thornton D.C., Tschudi M., Twohy C., Wetzel M., van Zanten M.C.: Dynamics and chemistry of marine stratocumulus: DYCOMS-II. Bull. Am. Meteorol. Soc. 84(5), 579 (2003)

    Article  Google Scholar 

  19. 19

    Stevens B., Moeng C.H., Ackerman A.S., Bretherton C.S., Zhu P., Chlond A., Müller F., De Roode S., Edwards J., Golaz J.: Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus. Mon. Weather Rev. 133, 1443–1462 (2005)

    Article  Google Scholar 

  20. 20

    Stevens B., Moeng C.H., Sullivan P.P.: Large-eddy simulations of radiatively driven convection: sensitivities to the representation of small scales. J. Atmos. Sci. 56, 3963–3984 (1999)

    Article  Google Scholar 

  21. 21

    Stevens B., Seifert A.: Understanding macrophysical outcomes of microphysical choices in simulations of shallow cumulus convection. J. Meteorol. Soc. Jap. 86(November), 143–162 (2008)

    Article  Google Scholar 

  22. 22

    Stevens D.E., Bell J.B., Almgren A.S., Beckner V.E., Rendleman C.A.: Small-scale processes and entrainment in a stratocumulus marine boundary layer. J. Atmos. Sci. 57(4), 567–581 (2000)

    Article  Google Scholar 

  23. 23

    Wunsch S.: Stochastic simulations of buoyancy-reversal experiments. Phys. Fluids 15(6), 1442–1456 (2003)

    Article  Google Scholar 

  24. 24

    Yamaguchi T., Randall D.A.: Large-eddy simulation of evaporatively driven entrainment in cloud-topped mixed layers. J. Atmos. Sci. 65(5), 1481–1504 (2008)

    Article  Google Scholar 

Download references


Financial support for this study was provided by the Deutsche Forschungsgemeinschaft within the SPP 1276 MetStröm and the Max Planck Society for the Advancement of Science. Computational resources have been provided by the German Climate Computing Centre, Hamburg.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Author information



Corresponding author

Correspondence to Eckhard Dietze.

Additional information

Communicated by R. Klein.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Dietze, E., Mellado, J.P., Stevens, B. et al. Study of low-order numerical effects in the two-dimensional cloud-top mixing layer. Theor. Comput. Fluid Dyn. 27, 239–251 (2013).

Download citation


  • Cloud-top mixing layer
  • Buoyancy reversal instability
  • Large-eddy simulation