Boundary-Layer Meteorology

, Volume 10, Issue 1, pp 35–54 | Cite as

The stability analysis of an inflexion-free velocity profile and its application to the night-time boundary layer in the atmosphere

  • D. Fua
  • F. Einaudi
  • D. P. Lalas
Article

Abstract

The characteristics of waves excited in a stratified shear flow with a velocity profile monotonically increasing above the ground are calculated numerically. It is shown that unstable modes exist when the Brunt-Väisälä frequency of the ambient atmosphere decreases sufficiently fast with height. Their growth rates as a function of the horizontal wavelength and the local Richardson number are given, and a comparison between them and experimental data obtained for the night-time-boundary layer of the Earth's atmosphere is carried out. Finally, the characteristics of the singular neutral modes that the system can support are presented.

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References

  1. Acton, F. S.: 1970, Numerical Methods that Work, Harper and Row, Publishers, New York.Google Scholar
  2. Bulirsch, R. and Stoer, J.: 1966, Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods, Num. Math. 8, 1–13.Google Scholar
  3. Case, K. M.: 1960, Stability of an Idealized Atmosphere. I: Discussion of Results, Phys. Fluids 3, 149–154.Google Scholar
  4. Chimonas, G.: 1970, The Extension of the Miles-Howard Theorem to Compressible Flows, J. Fluid Mech. 43, 833–836.Google Scholar
  5. Chimonas, G.: 1972, The Stability of a Coupled Wave-Turbulence System in a Parallel Shear Flow, Boundary-Layer Meteorol. 2, 444–452.Google Scholar
  6. Chimonas, G.: 1974, Considerations of the Stability of Certain Heterogeneous Shear Flows Including Some Inflexion-free Profiles, J. Fluid Mech. 65, 65–69.Google Scholar
  7. Einaudi, F. and Lalas, D. P.: 1976, The Effect of Boundaries on the Stability of Inviscid Stratified Shear Flows, accepted for publication in J. Appl. Mech. Google Scholar
  8. Emmanuel, C. B., Bean, B. R., McAllister, L. G., and Pollard, J. R.: 1972, Observations of Helmholtz Waves in the Lower Atmosphere with an Acoustic Sounder, J. Atmos. Sci. 29, 886–892.Google Scholar
  9. Gage, K. S. and Reid, W. H.: 1968, The Stability of Thermally Stratified Plane Poiseuille Flow, J. Fluid Mech. 33, 21–32.Google Scholar
  10. Gage, K. S.: 1971, The Effect of Stable Thermal Stratification on the Stability of Viscous Parallel Flows, J. Fluid Mech. 47, 1–20.Google Scholar
  11. Hall, F. F. and Owens, E. J.: 1975, ‘Atmospheric Acoustic Echo Soundings Investigations at the South Pole’, Proc. of the Workshop on Polar Meteorology, National Science Foundation, University of Nevada, May, 1975.Google Scholar
  12. Hines, C. O.: 1960, Internal Atmospheric Gravity Waves at Ionospheric Heights, Can. J. Phys. 38, 1441–1481.Google Scholar
  13. Hines, C. O.: 1974, The Upper Atmosphere in Motion, Geophysical Monograph 18, American Geophysical Union, Washington, D. C., pp. 248–328.Google Scholar
  14. Hooke, W. H., Hall, F. F., and Gossard, E. E.: 1973, Observed Generation of an Atmospheric Gravity Wave by Shear Instability in the Mean Flow of the Planetary Boundary Layer, Boundary-Layer Meteorol. 5, 29–41.Google Scholar
  15. Howard, L. N.: 1961, Note on a Paper of John W. Miles, J. Fluid Mech. 10, 509–512.Google Scholar
  16. Hull, T. E., Enright, W. H., Fellen, B. M. and Sedgwick, A. E.: 1972, Comparing Numerical Methods for Ordinary Differential Equations, SIAM J. Numer. Anal. 9, 603–637.Google Scholar
  17. Ince, E. L.: 1956, Ordinary Differential Equations, Dover Publications, Inc., New York.Google Scholar
  18. Jones, W. L.: 1968, Reflexion and Stability of Waves in Stably Stratified Fluids with Shear Flow: A Numerical Study, J. Fluid Mech. 34, 609–624.Google Scholar
  19. Lalas, D. P. and Einaudi, F.: 1975, On the Characteristics of Gravity Waves Generated by Atmospheric Shear Layers, submitted for publication.Google Scholar
  20. Maslowe, S. A. and Thompson, J. M.: 1971, Stability of a Stratified Free Shear Layer, Phys. Fluids 14, 453–458.Google Scholar
  21. Merrill, J. T.: 1975, ‘Observational Study of Shear Instability Events in the Planetary Boundary Layer’, 55th Annual Meeting Amer. Meteorol. Soc., Denver, Colo., June 20–23.Google Scholar
  22. Miles, J. W.: 1961, On the Stability of Heterogeneous Shear Flows, J. Fluid Mech. 10, 496–508.Google Scholar
  23. Miles, J. W.: 1963, On the Stability of Heterogeneous Shear Flows. Part 2, J. Fluid Mech. 16, 209–227.Google Scholar
  24. Miles, J. W.: 1967, Internal Waves in a Continuously Stratified Atmosphere or Ocean, J. Fluid Mech. 28, 305–310.Google Scholar
  25. Rayleigh, Lord: 1880, On the Stability, or Instability, of Certain Fluid Motions, Proc. London Math. Soc. 11, 57–70. (See also: Rayleigh, Lord, 1964, Scientific Papers 1, Dover Publications, pp. 474–487.)Google Scholar
  26. Schlichting, H.: 1935, Turbulenz bei Wärmeschichtung, Zeitschrift Angew. Math. Mech. 15, 313–338. (See also: ‘Turbulence and Heat Stratification’, Tech. Mem. 1262, National Advisory Committee for Aeronautics, Washington, D.C., 1950.)Google Scholar
  27. Stankov, B.: 1975, ‘Study of the Nocturnal Boundary Layer’, 55th Annual Meeting Amer. Meteorol. Soc., Denver, Colo., June 20–23.Google Scholar
  28. Synge, J. L.: 1933, The Stability of Heterogeneous Liquids, Trans. Royal Soc. Canada 27, 1–18.Google Scholar
  29. Thorpe, S. A.: 1969a, Experiments on the Stability of Stratified Shear Flows, Radio Science 4, 1327–1331.Google Scholar
  30. Thorpe, S. A.: 1969b, Neutral Eigensolutions of the Stability Equation for Stratified Shear Flow, J. Fluid Mech. 36, 673–683.Google Scholar
  31. Wagner, K. K.: Inflection Point Instability within an Inversion Layer, Atmospheric Research Laboratory Report, University of Oklahoma, Norman, Oklahoma, May 1975.Google Scholar
  32. Yih, C.-S.: 1970, Stability of and Waves in Stratified Flows, 8th Symposium on Hydrodynamics in the Ocean Environment, paper ARC-179, pp. 219–237.Google Scholar

Copyright information

© D. Reidel Publishing Company 1976

Authors and Affiliations

  • D. Fua
    • 1
  • F. Einaudi
    • 2
  • D. P. Lalas
    • 3
  1. 1.Cooperative Institute for Research in Environmental SciencesUniversity of Colorado/NOAABoulderUSA
  2. 2.Cooperative Institute for Research in Environmental SciencesUniversity of Colorado/NOAA and Aeronomy Laboratory, NOAABoulderUSA
  3. 3.Dept. of Mechanical Engineering SciencesWayne State UniversityDetroitUSA

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