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Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 1. Linear and nonlinear instabilities

Published online by Cambridge University Press:  08 January 2008

XUESONG WU  and
JING ZHANG
XUESONG WU
Affiliation:
Department of Mathematics, Imperial College London London SW7 2AZ, UK Department of Mechanics, Tianjin University, China
JING ZHANG
Affiliation:
Department of Mechanics, Tianjin University, China
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Abstract

In this paper, we consider a viscous instability of a stratified boundary layer that is a form of the familiar Tollmien–Schlichting (T-S) waves modified by a stable density stratification. As with the usual T-S waves, the triple-deck formalism was employed to provide a self-consistent description of linear and nonlinear instability properties at asymptotically large Reynolds numbers. The effect of stratification on the temporal and spatial linear growth rates is studied. It is found that stratification reduces the maximum spatial growth rate, but enhances the maximum temporal growth rate. This viscous instability may offer a possible alternative explanation for the origin of certain long atmospheric waves, whose characteristics are not well predicted by inviscid instabilities. In the high-frequency limit, the nonlinear evolution of the disturbances is shown to be governed by a nonlinear amplitude equation, which is an extension of the well-known Benjamin–Davis–Ono equation. Numerical solutions indicate that as a spatially isolated disturbance evolves, it radiates a beam of long gravity waves, and meanwhile small-scale ripples develop on its front to form a well-defined wavepacket. It is also shown that for jet-like velocity profiles, the standard triple-deck theory must be adjusted to account for both the displacement and transverse pressure variation induced by the inviscid flow in the main layer. The nonlinear evolution of high-frequency disturbances is governed by a mixed KdV–Benjamin–Davis–Ono equation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 379 - 408
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.CrossRefGoogle Scholar
Benney, D. J. 1966 Long nonlinear waves in fluid flows. J. Math. Phys. 45, 52–63.CrossRefGoogle Scholar
Caughey, S. J. & Reading, C. J. 1975 An observation of waves and turbulence in the earth's boundary layer. Boundary-Layer Met. 9, 279–296.CrossRefGoogle Scholar
Chanin, M.-L. & Haucheccorne, A. 1981 Lidar observation of gravity and tidal waves in the stratosphere and mesosphere. J. Geophys. Res. 86, 9715–9721.CrossRefGoogle Scholar
Chimonas, G. 1970 The extension of the Miles–Howard theorem to compressible fluids. J. Fluid Mech. 43, 833–836.CrossRefGoogle Scholar
Chimonas, G. 1974 Consideration of the stability of certain heterogenrous shear flows inclduing some inflexion-free profiles. J. Fluid Mech. 65, 65–69.CrossRefGoogle Scholar
Chimonas, G. 1993 Surface drag instabilities in the atmospheric boundary layer. J. Atmos. Sci. 50, 1914–1924.2.0.CO;2>CrossRefGoogle Scholar
Chimonas, G. 1995 Long-wavelength gravity instabilities: a comparison of the Geffrey's drag mechanism with the shear instability. J. Atmos. Sci. 52, 191–195.2.0.CO;2>CrossRefGoogle Scholar
Chimonas, G. 2002 On internal gravity waves associated with the stable boundary layer. Boundary-Layer Met. 102, 139–155.CrossRefGoogle Scholar
Chimonas, G. & Grant, J. R. 1984 Shear excitation of gravity waves. Part I: Modes of a two-scale atmosphere. J. Atmos. Sci. 41, 2269–2277.2.0.CO;2>CrossRefGoogle Scholar
Christie, D. R. 1989 Long nonlinear waves in the lower atmosphere. J. Atmos. Sci. 46, 1462–1491.2.0.CO;2>CrossRefGoogle Scholar
Coulter, R. L. & Doran, J. C. 2002 Spatial and temporal occurrence of intermittent turbulence during CASES-99. Boundary-Layer Met. 105, 329–349.CrossRefGoogle Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.CrossRefGoogle Scholar
Davis, P. A. & Peltier, W. R. 1976 Resonant parallel shear instability in the stratified planetary boundary layer. J. Atmos. Sci. 33, 1287–1300.2.0.CO;2>CrossRefGoogle Scholar
Davis, P. A. & Peltier, W. R. 1977 Effects of dissipation on parallel shear instability near the ground. J. Atmos. Sci. 34, 1868–1884.2.0.CO;2>CrossRefGoogle Scholar
Davis, P. A. & Peltier, W. R. 1979 Some characteristics of the Kelvin–Helmholtz and resonant overreflection modes of shear flow instability and of their interaction through vortex pairing. J. Atmos. Sci. 36, 2394–2412.2.0.CO;2>CrossRefGoogle Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluids. Adv.\ Appl. Mech. 9, 1–89.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Einaudi, F., Bedard, A. J. Jr & Finnigan, J. J. 1989 A climatology of gravity waves and other coherent disturbances at the Boulder atmospheric observatory during March–April 1984. J. Atmos. Sci. 46, 303–329.2.0.CO;2>CrossRefGoogle Scholar
Eymard, L. & Weill, A. 1979 A study of gravity waves in the planetary boundary layer by acoustic sounding. Boundary-Layer Met. 17, 231–245.CrossRefGoogle Scholar
Fabrikant, A. L. & Stepanyants, Yu. A. 1998 Propagation of waves in shear flows. World Scientific.CrossRefGoogle Scholar
Gear, J. A. & Grimshaw, R. 1983 A second-order theory for solitary waves in shallow fluids. Phys. Fluids 26, 14–29.CrossRefGoogle Scholar
Goldstein, S. 1931 On the stability of superposed streams of fluid of different densities. Proc. R. Soc. Lond. A 132, 524–548.Google Scholar
Gossard, E. E. & Hooke, W. H. 1975 Waves in the Atmosphere. Elsevier.Google Scholar
Grimshaw, R. 1981 A second-order theory for solitary waves in deep fluids. Phys. Fluids 24, 1611–1618.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
Lalas, D. P. & Einaudi, F. 1976 On the characteristics of gravity waves generated by atmospheric shear layer. J. Atmos. Sci. 33, 1248–1259.2.0.CO;2>CrossRefGoogle Scholar
Lalas, D. P., Einaudi, F. & Fua, D. 1976 The destabilizing effect of the ground on Kelvin–Helmholtz waves in the atmosphere. J. Atmos. Sci. 33, 59–69.2.0.CO;2>CrossRefGoogle Scholar
Lighthill, D. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lilly, D. K. & Zipser, E. J. 1972 The front range windstorm of 11 January 1972. A meteorological narrative. Weatherwise 25, 56–63.CrossRefGoogle Scholar
Lindzen, R. S. 1981 Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res. 86, 9707–9714.CrossRefGoogle Scholar
Maslowe, S. A. & Redekopp, L. G. 1980 Long nonlinear waves in stratifiled shear flows. J. Fluid Mech. 101, 321–348.CrossRefGoogle Scholar
Mastrantonio, G., Einaudi, F., Fua, D. & Lalas, D. P. 1976 Generation of gravity waves by jet streams in the atmosphere. J. Atmos. Sci. 33, 1730–1738.2.0.CO;2>CrossRefGoogle Scholar
Merrill, J. T. 1977 Observational and theoretical study of shear instability in the airflow near the ground. J. Atmos. Sci. 34, 911–921.2.0.CO;2>CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Mureithi, E. W. 1997 Effects of thermal bouyancy on the stability properties of boundary-layer flows. PhD thesis, University of New South Wales, Austrialia.Google Scholar
Mureithi, E. W., Denier, J. P. & Stott, J. A. K. 1997 The effect of buoyancy on upper branch Tollmien–Schlichting waves. IMA J. Appl. Maths 58, 19–50.CrossRefGoogle Scholar
Nakmura, R. & Mahrt, L. 2005 A study of intermittent turbulence with CASES-99 tower measurements. Boundary-Layer Met. 114, 367–387.Google Scholar
Newsom, R. K. & Banta, R. W. 2003 Shear-flow instability in the stable boundary layer as observed by Doppler lidar during CASES-99. J. Atmos. Sci. 60, 16–33.2.0.CO;2>CrossRefGoogle Scholar
Ohya, Y. & Uchida, T. 2003 Turbulence structure of stable boundary layers with a linear temperature profile. Boundary-Layer Met. 108, 19–38.CrossRefGoogle Scholar
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 1082–1091.CrossRefGoogle Scholar
Queney, P. 1948 The problem of air flow over mountains: a summary of theoretical studies. Bull. Am. Met. Soc. 29, 16–26.CrossRefGoogle Scholar
Ralph, F. M., Neiman, P. J. & Levinson, D. 1997 Lidar observations of a breaking mountain wave associated with extreme turbulence. Geophys. Res. Lett. 24, 663–666.CrossRefGoogle Scholar
Romanova, N. N. 1981 Generalization of the Benjamin–Ono equation for a weakly stratified atmosphere. Izv. Atmos. Ocean. Phys. 17, 98–101.Google Scholar
Romanova, N. N. 1984 Long nonlinear waves on layers having large wind velocity gradient. Izv. Atmos. Ocean. Phys. 20, 452–456.Google Scholar
Rottman, J. W. & Einaudi, F. 1993 Solitary waves in the atmosphere. J. Atmos. Sci. 50, 2116–2136.2.0.CO;2>CrossRefGoogle Scholar
Rottman, J. W. & Grimshaw, R. 2002 Environmental Stratified Flows. Kluwer.Google Scholar
Scorer, R. S. 1949 Theory of waves in the lee of mountains. Q. J. R. Met. Soc. 75, 41–56.CrossRefGoogle Scholar
Smith, F. T. 1973 Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803–824.CrossRefGoogle Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 336, 91–109.Google Scholar
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized short-scaled disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Smith, F. T. & Duck, P. W. 1977 Separation of jets or thermal boundary layers from a wall. Q. J. Mech. Appl. Maths 30, 145–156.CrossRefGoogle Scholar
Smith, R. B., Skubis, S., Doyle, J. D., Broad, A. S., Kiemle, C. & Volkert, H. 2002 Mountain waves over Mont Blanc: influence of a stagnant boundary layer. J. Atmos. Sci. 59, 2073–2092.2.0.CO;2>CrossRefGoogle Scholar
Smith, R. K. 1988 Travelling waves and bores in the lower atmosphere: the ‘morning glory’ and the related phenomenon. Earth-Sci. Rev. 25, 267–290.CrossRefGoogle Scholar
Smith, S. A. 2003 Observations and simulations of the 8 November MAP mountain wave case. Q. J. R. Met. Soc. 128, 1–21.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.CrossRefGoogle Scholar
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer.CrossRefGoogle Scholar
Sun, J., Burns, S. P., Lenschow, D. H., Banta, R., Newsom, R., Coulter, R., Frasier, S., Ince, T., Nappo, C., Cuxart, J., Blumen, W., Lee, X. & Hu, X. 2002 Intermittent turbulence associated with a density current passage in the stable boundary layer. Boundary-Layer Met. 105, 199–219.CrossRefGoogle Scholar
Sun, J., Lenschow, D. H., Burns, S. P., Newsom, R., Coulter, R., Frasier, S., Ince, T., Nappo, C., Balsley, R. B., Jensen, M., Mahrt, L., Miller, D. & Skelly, B. 2004 Atmospheric disturbances that generate intermittent turbulence in nocturnal boundary layers. Boundary-Layer Met. 110, 255–279.CrossRefGoogle Scholar
Sykes, R. I. 1978 Stratified effects in boundary layer flow over hills. Proc. R. Soc. Lond. A 361, 225–243.Google Scholar
Sykes, R. I. 1980 On three-dimensional boundary layer flow over surface irregularities. Proc. R. Soc. Lond. A 373, 311–329.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluids. Proc. R. Soc. Lond. A 132, 499–523.Google Scholar
Thorpe, S. A. 1969 Neutral eigensolutions of the stability equation for stratified shear flow. J. Fluid Mech. 36, 673–683.CrossRefGoogle Scholar
Voronovich, V. V., Shrira, V. I. & Stepanyants, Yu. A. 1998 Two-dimensional models for nonlinear vorticity waves in shear flows. Stud. Appl. Maths 100, 1–32.CrossRefGoogle Scholar
Wood, N. 2000 Wind flow over complex terrain: a historical perspective and the prospect for large-eddy modelling. Boundary-Layer Met. 96, 11–32.CrossRefGoogle Scholar
Wu, X. & Zhang, J. 2008 Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 2. Coupling with internal gravity waves via topography. J. Fluid Mech. 595, 409–433.CrossRefGoogle Scholar
Wurtele, M. G. 1970 Meteorological conditions surrounding the paradise crash of 1 March 1964. J. Appl. Met. 9, 787–795.2.0.CO;2>CrossRefGoogle Scholar