Abstract
A simple model is deduced for the surface layer of a convective boundary layer for zero mean wind velocity over homogeneous rough ground. The model assumes large-scale convective circulation driven by surface heat flux with a flow pattern as it would be obtained by conditional ensemble averages. The surface layer is defined here such that in this layer horizontal motions dominate relative to vertical components. The model is derived from momentum and heat balances for the surface layer together with closures based on the Monin-Obukhov theory. The motion in the surface layer is driven by horizontal gradients of hydrostatic pressure. The balances account for turbulent fluxes at the surface and fluxes by convective motions to the mixed layer. The latter are the dominant ones. The model contains effectively two empirical coefficients which are determined such that the model's predictions agree with previous experimental results for the horizontal turbulent velocity fluctuations and the temperature fluctuations. The model quantitatively predicts the decrease of the minimum friction velocity and the increase of the temperature difference between the mixed layer and the ground with increasing values of the boundary layer/roughness height ratio. The heat transfer relationship can be expressed in terms of the common Nusselt and Rayleigh numbers, Nu and Ra, as Nu ~ Ra% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca% aIXaaabaGaaGOmaaaaaaa!3779!\[{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\]. Previous results of the form Nu ~ Ra% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca% aIXaaabaGaaG4maaaaaaa!377A!\[{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}\] are shown to be restricted to Rayleigh-numbers less than a certain value which depends on the boundary layer/roughness height ratio.
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References
Businger, J. A.: 1973a, ‘A Note on Free Convection’, Boundary-Layer Meteorol. 4, 323–326.
Businger, J. A.: 1973b, ‘Turbulent Transfer in the Atmospheric Surface Layer’, in D. A. Haugen (ed.), Workshop on Micrometeorology, Amer. Meteor. Soc., Boston, pp. 67–100.
Businger, J. A., Wyngaard, J. C., Izumi, Y., and Bradley, E. F.: 1971, ‘Flux-Profile Relationships in the Atmospheric Surface Layer’, J. Atmos. Sci. 28, 181–189.
Caughey, S. J.: 1982, ‘Observed Characteristics of the Atmospheric Boundary Layer’, in F. T. M. Nieuwstadt and H. van Dop (ed.), Atmospheric Turbulence and Air Pollution Modelling, D. Reidel Publ., Dordrecht, pp. 107–158.
Deardorff, J. W.: 1970, ‘Convective Velocity and Temperature Scales for the Unstable Planetary Boundary Layer and for Rayleigh Convection’, J. Atmos. Sci. 27, 1211–1213.
Deardorff, J. W.: 1972, ‘Parameterization of the Planetary Boundary Layer for Use in General Circulation Models’, Monthly Weather Rev. 100, 93–105.
Deardorff, J. W. and Willis, G. E.: 1985, ‘Further Results from a Laboratory Model of the Convective Planetary Boundary Layer’, Boundary-Layer Meteorol. 32, 205–236.
Dyer, A. J.: 1974, ‘A Review of Flux-Profile Relationships’, Boundary-Layer Meteorol. 7, 363–372.
Globe, S. and Dropkin, D.: 1959, ‘Natural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Below’, J. Heat Transfer (ASME) 81, 24–28.
Greenhut, G. K. and Khalsa, S. J. S.: 1987, ‘Convective Elements in the Marine Atmospheric Boundary Layer. Part I: Conditional Sampling Statistics’, J. Clim. Appl. Meteorol. 26, 813–822.
Haugen, D. A., Kaimal, J. C., and Bradley, E. F.: 1971, ‘An Experimental Study of Reynolds Stress and Heat Flux in the Atmospheric Surface Layer’, Quart. J. Roy. Meteorol. Soc. 97, 168–180.
Kaimal, J. C., Wyngaard, J. C., Haugen, D. A., Cote, O. R., Izumi, Y., Caughey, S. J., and Readings, C. J.: 1976, ‘Turbulence Structure in the Convective Boundary Layer’, J. Atmos. Sci. 33, 2152–2169.
Moeng, C.-H. and Wyngaard, J. C.: 1986, ‘An Analysis of Closures for Pressure-Scalar Covariances in the Convective Boundary Layer’, J. Atmos. Sci. 43, 2499–2513.
Monin, A. S. and Yaglom, A. M.: 1971, Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press, Cambridge, Mass., Vol. 1.
Monin, A. S. and Zilitinkevich, S. S.: 1969, ‘On Description of Micro- and Meso-scale Phenomena in Numerical Models of the Atmosphere’, WMO-IUGG Symposium on Numerical Weather Forecasting Tokyo, 26 Nov.–4 Dec. 1968, Techn. Rep. Japan Meteorol. Agency No. 67, I.105–I.121.
Panofsky, H. A. and Dutton, J. A.: 1984, Atmospheric Turbulence, J. Wiley, New York, 397 pp.
Paulson, C. A.: 1970, ‘The Mathematical Representation of Wind Speed and Temperature Profiles in the Unstable Atmospheric Surface Layer’, J. Appl. Meteorol. 9, 857–861.
Prandtl, L.: 1932, ‘Meteorologische Anwendung der Strömungslehre’, Beitr. Phys. fr. Atmosph. 19, 188–202.
Priestley, C. H. B.: 1954, ‘Convection from a Large Horizontal Surface’, Australian J. Phys. 7, 176–201.
Schmid, H. and Schumann, U.: 1989, ‘ ‘ Coherent Structure of the Convective Boundary Layer Derived from Large-Eddy Simulations’, DFVLR-IB-553-2/88 ’, submitted to J. Fluid Mech.
Schumann, U., Hauf, T., Höller, H., Schmidt, H., and Volkert, H.: 1987, ‘A Meso scale Model for the Simulation of Turbulence, Clouds and Flow over Mountains: Formulation and Validation Examples’, Beitr. Phys. Atmosph. 60, 413–446.
Tennekes, H.: 1973, ‘Similarity Laws and Scale Relations in Planetary Boundary Layers’, in D. A. Haugen (ed.), Workshop on Micrometeorology, Amer. Meteor. Soc., Boston, pp. 177–216.
Townsend, A. A.: 1964, ‘Natural Convection in Water over an Ice Surface’, Quart. J. Roy. Meteorol. Soc. 90, 248–259.
Viswanadham, Y.: 1982, ‘Examination of the Empirical Flux-Profile Models in the Atmospheric Surface Boundary Layer’, Boundary-Layer Meteorol. 22, 61–77.
Wyngaard, J. C.: 1985, ‘Structure of the Planetary Boundary Layer and Implications for its Modeling’, J. Clim. Appl. Meteorol. 24, 1131–1142.
Wyngaard, J. C., Cote, O. R., and Izumi, Y.: 1971, ‘Local Free Convection, Similarity, and the Budgets of Shear Stress and Heat Flux’, J. Atmos. Sci. 28, 1171–1182.
Wier, M. and Römer, L.: 1987, ‘Experimentelle Untersuchung von stabil und instabil geschichteten turbulenten Plattengrenzschichten mit Bodenrauhigkeit’, Z. Flugwiss. Weltraumforsch. 11, 78–86.
Yaglom, A. M.: 1977, ‘Comments on Wind and Temperature Flux-Profile Relationships’, Boundary-Layer Meteorol. 11, 89–102.
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Schumann, U. Minimum friction velocity and heat transfer in the rough surface layer of a convective boundary layer. Boundary-Layer Meteorol 44, 311–326 (1988). https://doi.org/10.1007/BF00123019
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DOI: https://doi.org/10.1007/BF00123019