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
Acton, F. S.: 1970, Numerical Methods that Work, Harper and Row, Publishers, New York.
Bulirsch, R. and Stoer, J.: 1966, Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods, Num. Math. 8, 1–13.
Case, K. M.: 1960, Stability of an Idealized Atmosphere. I: Discussion of Results, Phys. Fluids 3, 149–154.
Chimonas, G.: 1970, The Extension of the Miles-Howard Theorem to Compressible Flows, J. Fluid Mech. 43, 833–836.
Chimonas, G.: 1972, The Stability of a Coupled Wave-Turbulence System in a Parallel Shear Flow, Boundary-Layer Meteorol. 2, 444–452.
Chimonas, G.: 1974, Considerations of the Stability of Certain Heterogeneous Shear Flows Including Some Inflexion-free Profiles, J. Fluid Mech. 65, 65–69.
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.
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.
Gage, K. S. and Reid, W. H.: 1968, The Stability of Thermally Stratified Plane Poiseuille Flow, J. Fluid Mech. 33, 21–32.
Gage, K. S.: 1971, The Effect of Stable Thermal Stratification on the Stability of Viscous Parallel Flows, J. Fluid Mech. 47, 1–20.
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.
Hines, C. O.: 1960, Internal Atmospheric Gravity Waves at Ionospheric Heights, Can. J. Phys. 38, 1441–1481.
Hines, C. O.: 1974, The Upper Atmosphere in Motion, Geophysical Monograph 18, American Geophysical Union, Washington, D. C., pp. 248–328.
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.
Howard, L. N.: 1961, Note on a Paper of John W. Miles, J. Fluid Mech. 10, 509–512.
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.
Ince, E. L.: 1956, Ordinary Differential Equations, Dover Publications, Inc., New York.
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.
Lalas, D. P. and Einaudi, F.: 1975, On the Characteristics of Gravity Waves Generated by Atmospheric Shear Layers, submitted for publication.
Maslowe, S. A. and Thompson, J. M.: 1971, Stability of a Stratified Free Shear Layer, Phys. Fluids 14, 453–458.
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.
Miles, J. W.: 1961, On the Stability of Heterogeneous Shear Flows, J. Fluid Mech. 10, 496–508.
Miles, J. W.: 1963, On the Stability of Heterogeneous Shear Flows. Part 2, J. Fluid Mech. 16, 209–227.
Miles, J. W.: 1967, Internal Waves in a Continuously Stratified Atmosphere or Ocean, J. Fluid Mech. 28, 305–310.
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.)
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.)
Stankov, B.: 1975, ‘Study of the Nocturnal Boundary Layer’, 55th Annual Meeting Amer. Meteorol. Soc., Denver, Colo., June 20–23.
Synge, J. L.: 1933, The Stability of Heterogeneous Liquids, Trans. Royal Soc. Canada 27, 1–18.
Thorpe, S. A.: 1969a, Experiments on the Stability of Stratified Shear Flows, Radio Science 4, 1327–1331.
Thorpe, S. A.: 1969b, Neutral Eigensolutions of the Stability Equation for Stratified Shear Flow, J. Fluid Mech. 36, 673–683.
Wagner, K. K.: Inflection Point Instability within an Inversion Layer, Atmospheric Research Laboratory Report, University of Oklahoma, Norman, Oklahoma, May 1975.
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.
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On a Fellowship from the National Research Council of Italy.
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Fua, D., Einaudi, F. & Lalas, D.P. The stability analysis of an inflexion-free velocity profile and its application to the night-time boundary layer in the atmosphere. Boundary-Layer Meteorol 10, 35–54 (1976). https://doi.org/10.1007/BF00218723
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DOI: https://doi.org/10.1007/BF00218723