Hans A. Panofsky’s Integral Similarity Function—At Fifty

DOI: 10.4236/acs.2013.34061   PDF   HTML     4,262 Downloads   7,434 Views   Citations


Fifty years ago, Hans A. Panofsky published a paper entitled Determination of stress from wind and temperature measurements. In his famous paper, he presented a new profile function for the mean horizontal wind speed under the condition of diabatic stratification that includes his integral similarity function. With his integral similarity function, he opened the door for Monin-Obukhov scaling in a wide range of micrometeorological and microclimatological applications. In a historic survey ranging from the sixties of the past century down to the present days, we present integral similarity functions for momentum, sensible heat, and water vapor for both unstable and stable stratification, where on the one hand free convection condition and on the other hand strongly stable stratification are addressed.

Share and Cite:

G. Kramm, D. Amaya, T. Foken and N. Mölders, "Hans A. Panofsky’s Integral Similarity Function—At Fifty," Atmospheric and Climate Sciences, Vol. 3 No. 4, 2013, pp. 581-594. doi: 10.4236/acs.2013.34061.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. A. Panofsky, “Determination of Stress from Wind and Temperature Measurements,” Quarterly Journal of the Royal Meteorological Society, Vol. 89, No. 379, 1963, pp. 85-94.
[2] A. S. Monin and A. M. Obukhov, “Osnovnye Zakonomernosti Turbulentnogo Peremesivanija v Prizemnom sloe Atmosfery,” Trudy Geofizicheskogo Instituta Akademiya Nauk SSSR, Vol. 24, 1954, pp. 163-187. (in Russian)
[3] O. Reynolds, “On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion,” Philosophical Transactions of the Royal Society, Vol. 186, 1895, pp. 123-164. http://dx.doi.org/10.1098/rsta.1895.0004
[4] T. Hesselberg, “Die Gesetze der Ausgeglichenen Atmospharischen Bewegungen,” Beitrage zur Physik der Atmosphare, Vol. 12, 1926, pp. 141-160. (in German)
[5] J. van Mieghem, “Atmospheric Energetics,” Clarendon Press, Oxford, 1973.
[6] G. Cox, “Basic Considerations,” In: G. Cox, Ed., Combustion Fundamentals of Fire, Academic Press, London, San Diego, New York, pp. 3-30.
[7] G. Kramm, R. Dlugi and D. H. Lenschow, “A Re-Evaluation of the Webb-Correction Using Density-Weighted Averages,” Journal of Hydrology, Vol. 166, No. 3-4, 1995, pp. 283-292.
[8] F. Herbert, “A Re-Evaluation of the Webb Correction Using Density-Weighted Averages—Comment,” Journal of Hydrology, Vol. 173, No. 1-4, 1995, pp. 343-344. http://dx.doi.org/10.1016/0022-1694(95)02867-O
[9] G. Kramm and F. X. Meixner, “On the Dispersion of Trace Species in the Atmospheric Boundary Layer: A Re-Formulation of the Governing Equations for the Turbulent Flow of the Compressible Atmosphere,” Tellus, Vol. 52A, 2000, pp. 500-522.
[10] A. S. Kowalski, “Exact Averaging of Atmospheric State and Flow Variables,” Journal of the Atmospheric Sciences, Vol. 69, No. 5, 2012, pp. 1750-1757. http://dx.doi.org/10.1175/JAS-D-11-0299.1
[11] E. Buckingham, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Physical Review, Vol. 4, No. 4, 1914, pp. 345-376. http://dx.doi.org/10.1103/PhysRev.4.345
[12] G. I. Barenblatt, “Similarity, Self-Similarity, and Intermediate Asymptotics,” Cambridge University Press, Cambridge, 1996.
[13] G. Kramm and F. Herbert, “Similarity Hypotheses for the Atmospheric Surface Layer Expressed by Dimensional π Invariants Analysis—A Review,” The Open Atmospheric Science Journal, Vol. 3, 2009, pp. 48-79.
[14] C. R. Stearns, “Determining Surface Roughness and Displacement Height,” Boundary-Layer Meteorology, Vol. 1, No. 1, 1970, pp. 102-111. http://dx.doi.org/10.1007/BF00193908
[15] A. K. Lo, “An Analytical-Empirical Method for Determining the Roughness Length and Zero-Plane Displacement,” Boundary-Layer Meteorology, Vol. 12, No. 2, 1977, pp. 141-151. http://dx.doi.org/ 10.1007/BF00121969
[16] F. Nieuwstadt, “The Computation of the Friction Velocity u* and the Temperature Scale T* from Temperature and Wind Velocity Profiles by Least-Square Methods,” Boundary-Layer Meteorology, Vol. 14, No. 2, 1978, pp. 235-246. http://dx.doi.org/10.1007/BF00122621
[17] G. Kramm, “The Estimation of the Surface Layer Parameters from Wind Velocity, Temperature and Humidity Profiles by Least Squares Methods,” Boundary-Layer Meteorology, Vol. 48, No. 3, 1989, pp. 315-327. http://dx.doi.org/10.1007/BF00158331
[18] G. Kramm, F. Herbert, K. Bernhardt, H. Müller, P. Werle, T. Foken and S. H. Richter, “Stability Functions for Momentum, Heat and Water Vapour and the Vertical Transport of TKE and Pressure Fluctuations Estimated from Measured Vertical Profiles of Wind Speed, Temperature, and Humidity,” Beitraege zur Physik der Atmosphaere, Vol. 69, 1996, pp. 463-475.
[19] C. A. Paulson, “The Mathematical Representation of Wind Speed and Temperature Profiles in the Unstable Atmospheric Surface Layer,” Journal of Applied Meteorology, Vol. 9, No. 6, 1970, pp. 857-861. http://dx.doi.org/10.1175/1520-0450(1970)009<0857:TMROWS>2.0.CO;2
[20] A. B. Kazansky and A. S. Monin, “Turbulence in the Inversion Layer near the Surface,” Izvestiya Akademii Nauk SSSR, Seriia Geofizicheskaia, Vol. 1, 1956, pp. 79-86.
[21] T. H. Ellison, “Turbulent Transport of Heat and Momentum from an Infinite Rough Plane,” Journal of Fluid Mechanics, Vol. 2, No. 5, 1957, pp. 456-466. http://dx.doi.org/10.1017/S0022112057000269
[22] G. Yamamoto, “Theory of Turbulent Transfer in Non-Neutral Conditions,” Journal of the Meteorological Society of Japan, Vol. 37, 1959, pp. 60-67.
[23] H. A. Panofsky, “An Alternative Derivation of the Diabatic Wind Profile,” Quarterly Journal of the Royal Meteorological Society, Vol. 87, No. 371, 1961, pp. 109-110. http://dx.doi.org/10.1002/qj. 49708737113
[24] W. D. Sellers, “Simplified Derivation of the Diabatic Wind Profile,” Journal of the Atmospheric Sciences, Vol. 19, No. 2, 1962, pp. 180-181. http://dx.doi.org/10.1175/1520-0469(1962)019 <0180:ASDOTD>2.0.CO;2
[25] J. A. Businger and A. M. Yaglom, “Introduction to Obkhov’s Paper ‘Turbulence in an Atmosphere with a Non-Uniform Temperature’,” Boundary-Layer Meteorology, Vol. 2, No. 1, 1971, pp. 3-6. http://dx.doi.org/10.1007/BF00718084
[26] T. Foken, “50 Years of Monin-Obukhov Similarity Theory,” Boundary-Layer Meteorology, Vol. 119, No. 3, 2006, pp. 431-447. http://dx.doi.org/10.1007/s10546-006-9048-6
[27] J. A. Businger, J. C. Wyngaard, Y. Izumi and E. F. Bradley, “Flux-Profile Relationships in the Atmospheric Surface Layer,” Journal of the Atmospheric Sciences, Vol. 28, No. 2, 1971, pp. 181-189. http://dx.doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2
[28] H. A. Panofsky and J. A. Dutton, “Atmospheric Turbulence,” John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore City, 1984.
[29] S. S. Zilitinkevic, “Effects of Humidity Stratification on Hydrostatic Stability,” Izvestiya, Atmospheric and Oceanic Physics, Vol. 2, 1966, pp. 655-658.
[30] K. Bernhardt, “Zur Interpretation der Monin-Obuchovschen Lange,” Meteorologische Zeitschrift, Vol. 4, 1995, pp. 81-82. (in German)
[31] A. J. Dyer, “A Review of Flux-Profile Relationships,” Boundary-Layer Meteorology, Vol. 7, No. 3, 1974, pp. 363-372. http://dx.doi.org/10.1007/BF00240838
[32] A. M. Yaglom, “Comments on Wind and Temperature Flux-Profile Relationships,” Boundary-Layer Meteorology, Vol. 11, No. 1, 1977, pp. 89-102. http://dx.doi.org/10.1007/BF00221826
[33] Z. Sorbjan, “Structure of the Atmospheric Boundary Layer,” Prentice Hall, Englewood Cliffs, 1989.
[34] J. R. Garratt, “The Atmospheric Boundary Layer,” Cambridge University Press, Cambridge, 1994.
[35] T. Foken, “Angewandte Meteorologie,” Springer, Berlin, Heidelberg, New York, 2003. (in German) http://dx.doi.org/10.1007/978-3-662-05743-8
[36] T. Foken, “Micrometeorology,” Springer, Berlin, Heidelberg, 2008.
[37] J. A. Businger, “Transfer of Momentum and Heat in the Planetary Boundary Layer,” Proceedings of the Symposium on Arctic Heat Budget and Atmospheric Circulation, Lake Arrowhead, The Rand Corporation, 31 January-4 February 1966, pp. 305-332.
[38] J. A. Businger, “A Note on the Businger-Dyer Profiles,” Boundary-Layer Meteorology, Vol. 42, No. 1-2, 1988, pp. 145-151. http://dx.doi.org/10.1007/BF00119880
[39] J. Pandolfo, “Wind and Temperature Profiles for a Constant Flux Boundary Layer in Lapse Conditions with a Variable Eddy Conductivity to Eddy Viscosity Ratio,” Journal of the Atmospheric Sciences, Vol. 23, No. 5, 1966, pp. 495-502. http://dx.doi.org/10.1175/1520-0469(1966)023< 0495:WATPFC>2.0.CO;2
[40] U. Hogstrom, “Non-Dimensional Wind and Temperature Profiles in the Atmospheric Surface Layer: A Re-Evaluation,” Boundary-Layer Meteorology, Vol. 42, No. 1-2, 1988, pp. 55-78.
[41] A. J. Dyer and B. B. Hicks, “Flux-Gradient Relationships in the Constant Flux Layer,” Quarterly Journal of the Royal Meteorological Society, Vol. 96, No. 410, 1970, pp. 715-721.
[42] A. J. Dyer and E. F. Bradley, “An Alternative Analysis of Flux-Gradient Relationships at the 1976 ITCE,” Boundary-Layer Meteorology, Vol. 22, No. 1, 1982, pp. 3-19. http://dx.doi.org/10.1007/BF00128053
[43] S. S. Zilitinkevic and D. V. Calikov, “Opredelenie Universal’Nych Profilej Skorosti vetra i Temperatury v Prizemnom sloe Atmosfery,” Izvestiya Akademii Nauk SSSR, Fizika Atmosfery i Okeana, Vol. 4, 1968, pp. 294-302. (in Russian)
[44] T. Foken and G. Skeib, “Profile Measurements in the Atmospheric Near-Surface Layer and the Use of Suitable Universal Functions for the Determination of the Turbulent Energy Exchange,” Boundary-Layer Meteorology, Vol. 25, No. 1, 1983, pp. 55-62. http://dx.doi.org/10.1007/BF00122097
[45] D. M. Carl, T. C. Tarbell and H. A. Panofsky, “Profiles of Wind and Temperature from Towers over Homogeneous Terrain,” Journal of the Atmospheric Sciences, Vol. 30, No. 5, 1973, pp. 788-794. http://dx.doi.org/10.1175/1520-0469(1973)030<0788:POWATF>2.0.CO;2
[46] A. S. Gavrilov and J. S. Petrov, “Ocenka Tocnosti Opredelenija Turbulentych Potokov po Standartnym Gidrometeorologiceskim Izmerenijam nad Morem,” Meteorologiya i Gidrologiya, Vol. 4, 1981, pp. 52-59. (in Russian)
[47] E. K. Webb, “Profile Relationships in the Super Adiabatic Surface Layer,” Quarterly Journal of the Royal Meteorological Society, Vol. 108, No. 457, 1982, pp. 661-688. http://dx.doi.org/ 10.1002/qj.49710845711
[48] D. A. Haugen, J. C. Kaimal and E.F. Bradley, “An Experimental Study of Reynolds Stress and Heat Flux in the Atmospheric Surface Layer,” Quarterly Journal of the Royal Meteorological Society, Vol. 97, No. 412, 1971, pp. 168-180. http://dx.doi.org/10.1002/qj.49709741204
[49] E. L. Andreas, K. J. Claffey, K. J. Jordan, C. W. Fairall, C. W. Guest, P. O. G. Persson and A. A. Grachev, “Evaluations of the von Kármán Constant in the Atmospheric Surface Layer,” Journal of Fluid Mechanics, Vol. 559, 2006, pp. 117-149. http://dx.doi.org/10.1017/S0022112006000164
[50] P. Frenzen and C. A. Vogel, “A Further Note ‘On the Magnitude and Apparent Range of Variation of the von Karman Constant’,” Boundary-Layer Meteorology, Vol. 73, No. 3, 1995, pp. 315-317. http://dx.doi.org/10.1007/BF00712700
[51] G. Kramm, “A Numerical Method for Determining the Dry Deposition of Atmospheric Trace Gases,” Boundary-Layer Meteorology, Vol. 48, No. 1-2, 1989, pp. 157-176. http://dx.doi.org/ 10.1007/BF00121788
[52] H. H. Lettau, “Wind and Temperature Profile Predictions for Diabatic Surface Layers Including Strong Inversion Cases,” Boundary-Layer Meteorology, Vol. 17, No. 4, 1979, pp. 443-464. http://dx.doi.org/10.1007/BF00118610
[53] L. Prandtl, “Meteorologische Anwendungen der Stromungslehre,” Beitraege zur Physik der Atmosphaere, (Bjerknes-Festband), Vol. 19, 1932, pp. 188-202. (in German)
[54] A. M. Obukhov, “Turbulentnost’ v Temperaturno-Neodnorodnoj Atmosphere,” Trudy Geofizicheskogo Instituta Akademiya Nauk SSSR, Vol. 1, 1946. (in Russian; a Translation into English Can Be Found in Boundary-Layer Meteorology, Vol. 2, 1971, p. 7).
[55] C. H. B. Priestley, “Turbulent Transfer in the Lower Atmosphere,” The University of Chicago Press, Chicago, 1959.
[56] J. L. Lumley and H. A. Panofsky, “The Structure of Atmospheric Turbulence,” Interscience Publishers (Wiley & Sons), New York, London, Sydney, 1964.
[57] M. A. Estoque, “Numerical Modeling of the Planetary Boundary Layer,” In: D. H. Haugen, Ed., Workshop on Micrometeorology, American Meteor Society, Boston, 1969, pp. 217-270.
[58] D. Amaya and G. Kramm, “A Modern Re-Evaluation of Free Convective Similarity Functions,” University of Alaska Fairbanks, Geophysical Institute, REU Summer Intern Report, 2012.
[59] B. A. Kader and A. M. Yaglom, “Mean Fields and Fluctuation Moments in Unstably Stratified Turbulent Boundary Layers,” Journal of Fluid Mechanics, Vol. 212, 1990, pp. 637-662.
[60] D. V. Calikov, “O Profilja vetra i Temperatury v Prizemnom sloe Atmosfery pri Ustojcivoj Stratifikacii,” Trudy GGO, Vol. 207, 1968, pp. 170-173. (in Russian)
[61] E. K. Webb, “Profile Relationships: The Log-Linear Range, and Extension to Strong Stability,” Quarterly Journal of the Royal Meteorological Society, Vol. 96, No. 407, 1970, pp. 67-90. http://dx.doi.org/10.1002/qj.49709640708
[62] N. Molders and G. Kramm, “A Case Study on Wintertime Inversions in Interior Alaska with WRF,” Atmospheric Research, Vol. 95, No. 2-3, 2010, pp. 314-332. http://dx.doi.org/10.1016/ j.atmosres.2009.06.002
[63] T. H. Ellison, “Laboratory Measurements of Turbulent Diffusion in Stratified Flows,” Journal of Geophysical Research, Vol. 67, No. 8, 1962, pp. 3029-3031. http://dx.doi.org/ 10.1029/JZ067i008p03029
[64] Y. Cheng and W. Brutsaert, “Flux-Profile Relationships for Wind Speed and Temperature in the Stable Atmospheric Boundary Layer,” Boundary-Layer Meteorology, Vol. 114, No. 3, 2005, pp. 519-538. http://dx.doi.org/10.1007/s10546-004-1425-4
[65] J.-F. Louis, “A Parametric Model of Vertical Eddy Fluxes in the Atmosphere,” Boundary-Layer Meteorology, Vol. 17, No. 2, 1979, pp. 187-202. http://dx.doi.org/10.1007/BF00117978
[66] A. C. M. Beljaars and A. A. M. Holtslag, “Flux Parameterization over Land Surfaces for Atmospheric Models,” Journal of Applied Meteorology, Vol. 30, No. 3, 1991, pp. 327-341.
[67] A. A. M. Holtslag and H. A. R. De Bruin, “Applied Modeling of the Nighttime Surface Energy Balance over Land,” Journal of Applied Meteorology, Vol. 27, No. 6, 1988, 689-704.
[68] D. Handorf, T. Foken and C Kottmeier, “The Stable Atmospheric Boundary Layer over an Antarctic Ice Sheet,” Boundary-Layer Meteorology, Vol. 91, No. 2, 1999, pp. 165-189.
[69] M. Okamoto and E. K. Webb, “The Temperature Fluctuations in Stable Stratification,” Quarterly Journal of the Royal Meteorological Society, Vol. 96, No. 410, 1970, pp. 591-600.
[70] J. Kondo, O. Kanechika and N. Yasuda, “Heat and Momentum Transfer under Strong Stability in the Atmospheric Surface Layer,” Journal of the Atmospheric Sciences, Vol. 35, No. 6, 1978, pp. 1012-1021. http://dx.doi.org/10.1175/1520-0469(1978)035<1012:HAMTUS>2.0.CO;2
[71] G. S. Poulos and S. P. Burns, “An Evaluation of Bulk Ri-Based Surface Layer Flux Formulas for Stable and Very Stable Conditions with Intermittent Turbulence,” Journal of the Atmospheric Sciences, Vol. 60, No. 20, 2003, pp. 2523-2537. http://dx.doi.org/10.1175/1520-0469(2003)060< 2523:AEOBRS>2.0.CO;2

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.