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Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 2. Coupling with internal gravity waves via topography

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

The aim of this paper is to show that the viscous shear instability identified in Part 1 is intrinsically coupled with internal gravity waves when a localized surface topography is present within a boundary layer. The coupling involves two aspects: receptivity and radiation. The former refers to excitation of shear instability modes by gravity waves, and the latter to emission of gravity waves by instability modes. Both physical processes are studied using triple-deck theory. In particular, the radiated gravity waves are found to produce a leading-order back action on the source, and this feedback effect, completely ignored in the acoustic analogy type of approach, is naturally taken into account by the triple-deck formalism. A by-product is that for certain incident angles, gravity waves are over-reflected by the boundary layer, i.e. the reflected waves are stronger than the incident waves.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 409 - 433
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
Acheson, D. J. 1976 On over-reflexion. J. Fluid Mech. 77, 433–472.CrossRefGoogle Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear layer. J. Fluid Mech. 27, 513–539.Google Scholar
Breeding, R. J. 1971 A nonlinear investigation of critical layers for the internal atmosphere gravity waves. J. Fluid Mech. 50, 545–563.Google Scholar
Chimonas, G. & Grant, J. R. 1984 Shear excitation of gravity waves. Part II: Upscale scattering from Kelvin–Helmoholtz waves. J. Atmos. Sci. 41, 2278–2288.Google Scholar
Clark, T. L., Hauf, T. & Kuettner, J. P. 1986 Convectively forced internal gravity waves: results from two-dimensional numerical experiments. Q. J. R. Met. Soc. 112, 899–925.Google Scholar
Coulter, R. L. & Doran, J. C. 2002 Spatial and temporal occurrence of intermittent turbulence during CASES-99. Boundary-Layer Met. 105, 329–349.Google 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
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.Google Scholar
Eltayeb, I. A. & McKenzie, J. F. 1975 Critical-layer behaviour and wave amplification of a gravity wave incident upon a shear layer. J. Fluid. Mech. 72, 661–671.CrossRefGoogle Scholar
Fritts, D. C. 1984 Shear excitation of atmospheric gravity wvaes. Part II: Nonlinear radiation from a shear layer. J. Atmos. Sci. 41, 524–537.Google Scholar
Fritts, D. C. 1989 A review of gravity saturation processes, effects, and variability in the middle atmosphere. Pure. Appl. Geophys. 130, 343–371.Google Scholar
Fritts, D. C. & Luo, Z. 1992 Gravity wave excitation by geostrophic adjustment of the jet stream. J. Atmos. Sci. 49, 681–697.Google Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509–529.CrossRefGoogle Scholar
Goldstein, M. E. & Hultgren, L. S. 1989 Boundary layer receptivity to long-wave free-stream turbulence. Annu. Rev. Fluid Mech. 21, 138.Google Scholar
Gossard, E. E. & Hooke, W. H. 1975 Waves in the Atmosphere. Elsevier.Google Scholar
Grimshaw, R. H. J. 1981 Resonant over-reflection of internal gravity waves from a thin shear layer. J. Fluid Mech. 109, 349–365.Google Scholar
Hauf, T. & Clark, T. L. 1989 Three-dimensional numerical experiments on convectively forced internal gravity waves. Q. J. R. Met. Soc. 115, 309–333.Google Scholar
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.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.Google Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically. Proc. R. Soc. Lond. A 211, 564–587.Google Scholar
Lindzen, R. S. 1981 Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res. 86, 9707–9714.CrossRefGoogle Scholar
Lindzen, R. S. & Barker, J. W. 1985 Instability and wave over-reflection in stably stratified shear flow. J. Fluid Mech. 151, 189–217.Google Scholar
Lott, F., Kelder, H. & Teitelbaum, H. 1992 A transition from Kelvin–Helmholtz instabilities to propagating instabilities. Phys. Fluids A 4 (9), 1990–1997.Google Scholar
McIntyre, M. E. & Weissman, M. A. 1978 On radiating instabilities and resonant overreflection. J. Atmos. Sci. 35, 1190–1196.Google 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.Google Scholar
Nakmura, R. & Mahrt, L. 2005 A study of intermittent turbulence with CASES-99 tower measurements. Boundary-Layer Met. 114, 367–387.Google Scholar
Ohya, Y. & Uchida, T. 2003 Turbulence structure of stable boundary layers with a linear temperature profile. Boundary-Layer Met. 108, 19–38.Google Scholar
Reinking, R. F., Frisch, A. S., Orr, B. W., Korn, D. L., Bissonnette, L. R. & Roy, G. 2003 Remote sensing observation of effects of moutain blocking on travelling gravity-shear waves and associated clouds. Boundary-Layer Met. 109, 255–284.CrossRefGoogle Scholar
Ruban, A. I. 1984 On the generation of Tollmien–Schlichting waves by sound. Fluid Dyn. 19, 709–716.Google Scholar
Scinocca, J. F. & Ford, R. 2000 The nonlinear forcing of large-scale internal gravity waves by stratified shear instability. J. Atmos. Sci. 57, 653–672.Google Scholar
Smith, R. B. 1979 The influence of mountains on the atmosphere. Adv. Geophys. 21, 87–230.Google Scholar
Stein, R. F. 1967 Generation of acoustic and gravity waves by turbulence in an isothermal stratified atmosphere. Solar Phys. 2, 385–432.CrossRefGoogle Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.Google 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.Google 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.Google 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
Wu, X. 2001 On local boundary-layer receptivity to vortical disturbances in the free stream. J. Fluid Mech. 449, 373–393.CrossRefGoogle Scholar
Wu, X. & Hogg, L. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 303–347.CrossRefGoogle Scholar
Wu, X. & Zhang, J. 2008 Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 1. Linear and nonlinear instabilities. J. Fluid Mech. 595, 379–408.CrossRefGoogle Scholar