Skip to main content

Cloud microphysical effects of turbulent mixing and entrainment


Turbulent mixing and entrainment at the boundary of a cloud is studied by means of direct numerical simulations that couple the Eulerian description of the turbulent velocity and water vapor fields with a Lagrangian ensemble of cloud water droplets that can grow and shrink by condensation and evaporation, respectively. The focus is on detailed analysis of the relaxation process of the droplet ensemble during the entrainment of subsaturated air, in particular the dependence on turbulence timescales, droplet number density, initial droplet radius and particle inertia. We find that the droplet evolution during the entrainment process is captured best by a phase relaxation time that is based on the droplet number density with respect to the entire simulation domain and the initial droplet radius. Even under conditions favoring homogeneous mixing, the probability density function of supersaturation at droplet locations exhibits initially strong negative skewness, consistent with droplets near the cloud boundary being suddenly mixed into clear air, but rapidly approaches a narrower, symmetric shape. The droplet size distribution, which is initialized as perfectly monodisperse, broadens and also becomes somewhat negatively skewed. Particle inertia and gravitational settling lead to a more rapid initial evaporation, but ultimately only to slight depletion of both tails of the droplet size distribution. The Reynolds number dependence of the mixing process remained weak over the parameter range studied, most probably due to the fact that the inhomogeneous mixing regime could not be fully accessed when phase relaxation times based on global number density are considered.

This is a preview of subscription content, access via your institution.


  1. Blyth A.M.: Entrainment in cumulus clouds. J. Appl. Meteorol. 32, 626–640 (1993)

    Article  Google Scholar 

  2. Shaw R.A.: Particle-turbulence interaction in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227 (2003)

    Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  4. Wang S., Golaz J.-C., Wang Q.: Effect of intense wind shear across the inversion on stratocumulus clouds. Geophys. Res. Lett. 35, L15814 (2008)

    Article  Google Scholar 

  5. Katzwinkel, J., Siebert, H., Shaw, R.A.: Observation of a self-limiting, shear-induced turbulent inversion layer above marine stratocumulus. Boundary Layer Meteorol. doi:10.1007/s10546-011-9683-4 (2011)

  6. Latham J., Reed R.L.: Laboratory studies of the effects of mixing on the evolution of cloud droplet spectra. Q. J. R. Meteorol. Soc. 103, 297–306 (1977)

    Article  Google Scholar 

  7. Baker M.B., Breidenthal R.E., Choularton T.W., Latham J.: The effects of turbulent mixing in clouds. J. Atmos. Sci. 41, 299–304 (1984)

    Article  Google Scholar 

  8. Jensen J.B., Baker M.B.: A simple model of droplet spectral evolution during turbulent mixing. J. Atmos. Sci. 46, 2812–2829 (1989)

    Article  Google Scholar 

  9. Lehmann K., Siebert H., Shaw R.A.: Homogeneous and inhomogeneous mixing in cumulus clouds: dependence on local turbulence structure. J. Atmos. Sci. 66, 3641–3659 (2009)

    Article  Google Scholar 

  10. Vaillancourt P.A., Yau M.K., Grabowski W.W.: Microscopic approach to cloud droplet growth by condensation. Part I: model description and results without turbulence. J. Atmos. Sci. 58, 1945–1964 (2001)

    Article  Google Scholar 

  11. Andrejczuk M., Grabowski W.W., Malinowski S.P., Smolarkiewicz P.K.: Numerical simulation of cloud-clear-air interfacial mixing. J. Atmos. Sci. 61, 1726–1739 (2004)

    Article  Google Scholar 

  12. Andrejczuk M., Grabowski W.W., Malinowski S.P., Smolarkiewicz P.K.: Numerical simulation of cloud-clear-air interfacial mixing: effects on cloud microphysics. J. Atmos. Sci. 63, 3204–3225 (2006)

    Article  Google Scholar 

  13. Lanotte A., Seminara A., Toschi F.: Cloud droplet growth by condensation in homogeneous isotropic turbulence. J. Atmos. Sci. 66, 1685–1697 (2009)

    Article  Google Scholar 

  14. Lamb D., Verlinde J.: Physics and Chemistry of Clouds. Cambridge Univ. Press, Cambridge (2011)

    Book  Google Scholar 

  15. Kostinski A.B.: Simple approximations for condensational growth. Environ. Res. Lett. 4, 015005 (2009)

    Article  Google Scholar 

  16. Rogers R.R., Yau M.K.: A Short Course in Cloud Physics. Butterworth-Heinemann, Woburn (1989)

    Google Scholar 

  17. Schumacher J., Sreenivasan K.R., Yakhot V.: Asymptotic exponents from low-Reynolds-number turbulent flows. New J. Phys. 9, 89 (2007)

    Article  Google Scholar 

  18. Eswaran V., Pope S.B.: An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16, 257–278 (1988)

    MATH  Article  Google Scholar 

  19. Schumacher J., Eckhardt B., Doering C.R.: Extreme vorticity growth in Navier-Stokes turbulence. Phys. Lett. A 374, 861–865 (2010)

    MATH  Article  Google Scholar 

  20. Kolmogorov A.N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  21. Knaepen B., Debliquy O., Carati D.: Direct numerical simulation and large-eddy simulation of a shear-free mixing layer. J. Fluid Mech. 414, 153–172 (2004)

    Article  Google Scholar 

  22. Andrejczuk M., Grabowski W.W., Malinowski S.P., Smolarkiewicz P.K.: Numerical simulation of cloud–clear air interfacial mixing: homogeneous versus inhomogeneous mixing. J. Atmos. Sci. 66, 2493–2500 (2009)

    Article  Google Scholar 

  23. Siebert H., Gerashchenko S., Gylfason A., Lehmann K., Collins L.R., Shaw R.A., Warhaft Z.: Towards understanding the role of turbulence on droplets in clouds: in situ and laboratory measurements. Atmos. Res. 97, 426–437 (2010)

    Article  Google Scholar 

  24. Steinfeld G., Raasch S., Markkanen T.: Footprints in homogeneously and heterogeneously driven boundary layers derived from a Lagrangian stochastic particle model embedded into large-eddy simulation. Boundary Layer Meteorol. 129, 225–248 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Bipin Kumar.

Additional information

Communicated by R. Klein.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kumar, B., Schumacher, J. & Shaw, R.A. Cloud microphysical effects of turbulent mixing and entrainment. Theor. Comput. Fluid Dyn. 27, 361–376 (2013).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Turbulent mixing
  • Entrainment
  • Cloud droplets
  • Phase relaxation