Editorial Type:
Article
Article Type:
Research Article

Asymptotic Analysis of Equilibrium in Radiation Fog

Binbin Zhou Science Applications International Corporation, Camp Springs, Maryland

Search for other papers by Binbin Zhou in
Current site
Google Scholar
PubMed Close
and
Brad S. Ferrier Science Applications International Corporation, Camp Springs, Maryland

Search for other papers by Brad S. Ferrier in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Jun 2008
DOI:
https://doi.org/10.1175/2007JAMC1685.1
Page(s):
1704–1722
Restricted access

Abstract

A vertical distribution formulation of liquid water content (LWC) for steady radiation fog was obtained and examined through the singular perturbation method. The asymptotic LWC distribution is a consequential balance among cooling, droplet gravitational settling, and turbulence in the liquid water budget of radiation fog. The cooling produces liquid water, which is depleted by turbulence near the surface. The influence of turbulence on the liquid water budget decreases with height and is more significant for shallow fogs than for deep fogs. The depth of the region of surface-induced turbulence can be characterized with a fog boundary layer (FBL). The behavior of the FBL bears some resemblance to the surface mixing layer in radiation fog. The characteristic depth of the FBL is thinner for weaker turbulence and stronger cooling, whereas if turbulence intensity increases or cooling rate decreases then the FBL will develop from the ground. The asymptotic formulation also reveals a critical turbulent exchange coefficient for radiation fog that defines the upper bound of turbulence intensity that a steady fog can withstand. The deeper a fog is, the stronger a turbulence intensity it can endure. The persistence condition for a steady fog can be parameterized by either the critical turbulent exchange coefficient or the characteristic depth of the FBL. If the turbulence intensity inside a fog is smaller than the turbulence threshold, the fog persists, whereas if the turbulence intensity exceeds the turbulence threshold or the characteristic depth of the FBL dominates the entire fog bank then the balance will be destroyed, leading to dissipation of the existing fog. The asymptotic formulation has a first-order approximation with respect to turbulence intensity. Verifications with numerical solutions and an observed fog event showed that it is more accurate for weak turbulence than for strong turbulence and that the computed LWC generally agrees with the observed LWC in magnitude.

Corresponding author address: Binbin Zhou, NCEP Environmental Modeling Center, 5200 Auth Road, Camp Springs, MD 20646. Email: binbin.zhou@noaa.gov

Abstract

A vertical distribution formulation of liquid water content (LWC) for steady radiation fog was obtained and examined through the singular perturbation method. The asymptotic LWC distribution is a consequential balance among cooling, droplet gravitational settling, and turbulence in the liquid water budget of radiation fog. The cooling produces liquid water, which is depleted by turbulence near the surface. The influence of turbulence on the liquid water budget decreases with height and is more significant for shallow fogs than for deep fogs. The depth of the region of surface-induced turbulence can be characterized with a fog boundary layer (FBL). The behavior of the FBL bears some resemblance to the surface mixing layer in radiation fog. The characteristic depth of the FBL is thinner for weaker turbulence and stronger cooling, whereas if turbulence intensity increases or cooling rate decreases then the FBL will develop from the ground. The asymptotic formulation also reveals a critical turbulent exchange coefficient for radiation fog that defines the upper bound of turbulence intensity that a steady fog can withstand. The deeper a fog is, the stronger a turbulence intensity it can endure. The persistence condition for a steady fog can be parameterized by either the critical turbulent exchange coefficient or the characteristic depth of the FBL. If the turbulence intensity inside a fog is smaller than the turbulence threshold, the fog persists, whereas if the turbulence intensity exceeds the turbulence threshold or the characteristic depth of the FBL dominates the entire fog bank then the balance will be destroyed, leading to dissipation of the existing fog. The asymptotic formulation has a first-order approximation with respect to turbulence intensity. Verifications with numerical solutions and an observed fog event showed that it is more accurate for weak turbulence than for strong turbulence and that the computed LWC generally agrees with the observed LWC in magnitude.

Corresponding author address: Binbin Zhou, NCEP Environmental Modeling Center, 5200 Auth Road, Camp Springs, MD 20646. Email: binbin.zhou@noaa.gov

Save
Cover Journal of Applied Meteorology and Climatology
Journal of Applied Meteorology and Climatology

  • Ballard, S. P., B. W. Golding, and R. N. B. Smith, 1991: Mesoscale experimental forecasts of the Haar of northeast Scotland. Mon. Wea. Rev., 119 , 2107–2123.

    • Search Google Scholar
    • Export Citation

  • Bergot, T., and D. Guedalia, 1994: Numerical forecasting of radiation fog. Part I: Numerical model and sensitivity tests. Mon. Wea. Rev., 122 , 1218–1230.

    • Search Google Scholar
    • Export Citation

  • Bergot, T., D. Carrer, J. Noilhan, and P. Bougeault, 2005: Improved site-specific numerical prediction of fog and low clouds: A feasibility study. Wea. Forecasting, 20 , 627–646.

    • Search Google Scholar
    • Export Citation

  • Blackadar, A. K., and H. Tennekes, 1968: Asymptotic similarity in neutral barotropic planetary boundary layers. J. Atmos. Sci., 25 , 1015–1020.

    • Search Google Scholar
    • Export Citation

  • Bott, A., U. Sievers, and W. Zdunkowski, 1990: A radiation fog model with a detailed treatment of the interaction between radiative transfer and fog microphysics. J. Atmos. Sci., 47 , 2153–2166.

    • Search Google Scholar
    • Export Citation

  • Brown, R., and W. T. Roach, 1976: The physics of radiation fog: II—A numerical study. Quart. J. Roy. Meteor. Soc., 102 , 335–354.

    • Search Google Scholar
    • Export Citation

  • Busch, N. E., 1973: On the mechanics of atmospheric turbulence. Workshop on Micrometeorology, D. A. Haugen, Ed., Amer. Meteor. Soc., 1–28.

    • Search Google Scholar
    • Export Citation

  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28 , 181–189.

    • Search Google Scholar
    • Export Citation

  • Choularton, T. W., G. Fullarton, J. Latham, C. S. Mill, M. H. Smith, and I. M. Stromberg, 1981: A field study of radiation fog in Meppen, West Germany. Quart. J. Roy. Meteor. Soc., 107 , 381–394.

    • Search Google Scholar
    • Export Citation

  • Croft, P. J., R. L. Pfost, J. M. Medlin, and G. A. Johnson, 1997: Fog forecasting for the southern region: A conceptual model approach. Wea. Forecasting, 12 , 545–556.

    • Search Google Scholar
    • Export Citation

  • Duynkerke, P. G., 1991: Radiation fog: A comparison of model simulation with detailed observations. Mon. Wea. Rev., 119 , 324–341.

    • Search Google Scholar
    • Export Citation

  • Duynkerke, P. G., 1999: Turbulence, radiation and fog in Dutch stable boundary layers. Bound.-Layer Meteor., 90 , 447–477.

  • Fitzjarrald, D. R., and G. G. Lala, 1989: Hudson Valley fog environments. J. Appl. Meteor., 28 , 1303–1328.

  • Fuzzi, S., and Coauthors, 1992: The Po Valley fog experiment 1989: An overview. Tellus, 44B , 448–468.

  • Gerber, H. E., 1981: Microstructure of a radiation fog. J. Atmos. Sci., 38 , 454–458.

  • Girard, E., and J-P. Blanchet, 2001: Simulation of Arctic diamond dust, ice fog, and thin stratus using an explicit aerosol–cloud–radiation model. J. Atmos. Sci., 58 , 1199–1221.

    • Search Google Scholar
    • Export Citation

  • Golding, B. W., 1993: A study of the influence of terrain on fog development. Mon. Wea. Rev., 121 , 2529–2541.

  • Guedalia, D., and T. Bergot, 1994: Numerical forecasting of radiation fog. Part II: A comparison of model simulation with several observed fog events. Mon. Wea. Rev., 122 , 1231–1246.

    • Search Google Scholar
    • Export Citation

  • Gultepe, I., M. D. Müller, and Z. Boybeyi, 2006: A new visibility parameterization for warm-fog applications in numerical weather prediction models. J. Appl. Meteor. Climatol., 45 , 1469–1480.

    • Search Google Scholar
    • Export Citation

  • Jiusto, J. E., 1981: Fog structure. Clouds: Their Formation, Optical Properties and Effects, P. V. Hobbs and A. Deepak, Eds., Academic Press, 187–239.

    • Search Google Scholar
    • Export Citation

  • Jiusto, J. E., and G. G. Lala, 1980: Radiation fog formation and dissipation. A case study. J. Rech. Atmos., 14 , 391–397.

  • Kelley, W., 1994: Book review: Introduction to the General Theory of Singular Perturbations. Bull. Amer. Math. Soc., 30 , 258–261.

    • Search Google Scholar
    • Export Citation

  • Kunkel, B. A., 1984: Parameterization of droplet terminal velocity and extinction coefficient in fog models. J. Climate Appl. Meteor., 23 , 34–41.

    • Search Google Scholar
    • Export Citation

  • Lala, G. G., E. Mandel, and J. E. Jiusto, 1975: A numerical evaluation of radiation fog variables. J. Atmos. Sci., 32 , 720–728.

  • Mellor, G. L., and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci., 31 , 1791–1806.

    • Search Google Scholar
    • Export Citation

  • Meyer, M. B., and G. G. Lala, 1990: Climatological aspects of radiation fog occurrence at Albany, New York. J. Climate, 3 , 577–586.

    • Search Google Scholar
    • Export Citation

  • Meyer, M. B., G. G. Lala, and J. E. Jiusto, 1986: FOG-82: A cooperative field study of radiation fog. Bull. Amer. Meteor. Soc., 67 , 825–832.

    • Search Google Scholar
    • Export Citation

  • Meyer, W. D., and G. V. Rao, 1999: Radiation fog prediction using a simple numerical model. Pure Appl. Geophys., 155 , 57–80.

  • Müller, M. D., M. Masbou, A. Bott, and Z. Janjic, 2005: Fog prediction in a 3D model with parameterized microphysics. Proc. Symp. on Nowcasting and Very Short Range Forecasting, Toulouse, France, World Weather Research Programme, 6.26. [Available online at http://www.meteo.fr/cic/wsn05/DVD/index.html.].

  • Musson-Genon, L., 1987: Numerical simulation of a fog event with a one-dimensional boundary layer model. Mon. Wea. Rev., 115 , 592–607.

    • Search Google Scholar
    • Export Citation

  • Nakanishi, M., 2000: Large-eddy simulation of radiation fog. Bound.-Layer Meteor., 94 , 461–493.

  • Oliver, D. A., W. S. Lewellen, and G. G. Williamson, 1978: The interaction between turbulent and radiative transport in the development of fog and low-level stratus. J. Atmos. Sci., 35 , 301–316.

    • Search Google Scholar
    • Export Citation

  • O’Malley Jr, R. E., 1974: Introduction to Singular Perturbations. Academic Press, 206 pp.

  • Pagowski, M., I. Gultepe, and P. King, 2004: Analysis and modeling of an extremely dense fog event in southern Ontario. J. Appl. Meteor., 43 , 3–16.

    • Search Google Scholar
    • Export Citation

  • Pickering, K. E., and J. E. Jiusto, 1978: Observations of the relationship between dew and radiation fog. J. Geophys. Res., 83 , 2430–2436.

    • Search Google Scholar
    • Export Citation

  • Pilié, R. J., E. J. Mack, W. C. Kocmond, W. J. Eadie, and C. W. Rogers, 1975: The life cycle of valley fog. Part II: Fog microphysics. J. Appl. Meteor., 14 , 347–363.

    • Search Google Scholar
    • Export Citation

  • Pinnick, R. G., D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, and S. G. Jennings, 1978: Vertical structure in atmospheric fog and haze and its effects on visible and infrared extinction. J. Atmos. Sci., 35 , 2020–2032.

    • Search Google Scholar
    • Export Citation

  • Prandtl, L., 1904: Ãœber Flüssigkeitsbewegung bei sehr kleiner Reibung (On fluid motion with very small friction). Vehr. III Int. Math. Kongr., Heidelburg, 484–491.

  • Roach, W. T., R. Brown, S. J. Caughey, J. A. Garland, and C. J. Readings, 1976: The physics of radiation fog. I: A field study. Quart. J. Roy. Meteor. Soc., 102 , 313–333.

    • Search Google Scholar
    • Export Citation

  • Seydel, R., 1988: From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis. Elsevier, 367 pp.

  • Stoelinga, M. T., and T. T. Warner, 1999: Nonhydrostatic, mesobeta-scale model simulations of cloud ceiling and visibility for an East Coast winter precipitation event. J. Appl. Meteor., 38 , 385–404.

    • Search Google Scholar
    • Export Citation

  • Stull, R. B., 1988: An. Introduction to Boundary Layer Meteorology, Kluwer Academic, 666 pp.

  • Taylor, G. I., 1917: The formation of fog and mist. Quart. J. Roy. Meteor. Soc., 43 , 241–268.

  • Tardif, R., 2004a: Characterizing fog occurrences in the northeastern United States using historical data. Preprints, 11th Conf. on Aviation, Range, and Aerospace, Hyannis, MA, Amer. Meteor. Soc., 5.1. [Available online at http://ams.confex.com/ams/pdfpapers/101782.pdf.].

  • Tardif, R., J. A. Cole, P. H. Herzegh, S. D. Landoit, R. M. Rasmussen, and M. L. Tryhane, 2004b: First observations of fog and low ceiling environments at the FAA northeast ceiling and visibility field site. Preprints, 11th Conf. on Aviation, Range, and Aerospace, Hyannis, MA, Amer. Meteor. Soc., 10.5. [Available online at http://ams.confex.com/ams/pdfpapers/81649.pdf.].

  • Thuman, W. C., and E. Robinson, 1954: Studies of Alaskan ice-fog particles. J. Meteor., 11 , 151–156.

  • Turton, J. D., and R. Brown, 1987: A comparison of a numerical model of radiation fog with detailed observations. Quart. J. Roy. Meteor. Soc., 113 , 37–54.

    • Search Google Scholar
    • Export Citation

  • Van Dyke, M., 1964: Perturbation Methods in Fluid Mechanics. Academic Press, 229 pp.

  • Von Glasow, R., and A. Bott, 1999: Interaction of radiation fog with tall vegetation. Atmos. Environ., 33 , 1333–1346.

  • Welch, R. M., M. G. Ravichandran, and S. K. Cox, 1986: Prediction of quasi-periodic oscillations in radiation fogs. Part I: Comparison of simple similarity approaches. J. Atmos. Sci., 43 , 633–651.

    • Search Google Scholar
    • Export Citation

  • Westcott, N., 2004: Environmental conditions associated with dense fog in Illinois. Preprints, 11th Conf. on Aviation, Range, and Aerospace, Hyannis, MA, Amer. Meteor. Soc., P10.4. [Available online at http://ams.confex.com/ams/pdfpapers/81513.pdf.].

  • Zdunkowski, W. G., and A. E. Barr, 1972: A radiative-conductive model for the prediction of radiation fog. Bound.-Layer Meteor., 3 , 152–177.

    • Search Google Scholar
    • Export Citation

  • Zhou, B., 1987: Numerical modeling of radiation fog (in Chinese). ACTA Meteor. Sin., 45 , 21–29.

  • Zhou, B., 2006: Asymptotic solution of liquid water content equation at equilibrium state of radiation fog. NCEP Office Note 450, 41 pp.

  • Zhou, B., J. Du, B. Ferrier, J. McQueen, and G. DiMego, 2007: Numerical forecast of fog—Central solutions. Preprints, 22nd Conf. on Weather Analysis and Forecasting and 18th Conf. on Numerical Weather Prediction, Park City, UT, Amer. Meteor. Soc. [Available online at http://ams.confex.com/ams/pdfpapers/123669.pdf.].

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 1598 751 33
PDF Downloads 662 106 9