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Reflection properties of internal gravity waves incident upon a hyperbolic tangent shear layer

Published online by Cambridge University Press:  20 April 2006

Cornelis A. Van Duin  and
Hennie Kelder
Cornelis A. Van Duin
Affiliation:
Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, the Netherlands Present address: Division of Geophysics, Royal Netherlands Meteorological Institute, 3730 AE De Bilt, the Netherlands
Hennie Kelder
Affiliation:
Division of Geophysics, Royal Netherlands Meteorological Institute, 3730 AE De Bilt, the Netherlands
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Abstract

The properties of reflection and transmission of internal gravity waves incident upon a shear layer containing a critical level are investigated. The shear layer is modelled by a hyperbolic tangent profile. In the Boussinesq approximation, the differential equation governing the propagation of these waves can then be transformed into Heun's equation. For large Richardson numbers this equation can be approximated by an equation that has solutions in terms of hypergeometric functions. For these values of the Richardson number the reflection coefficient proves to be strongly dependent on the place of the critical level in the shear flow. If the Doppler-shifted frequency is an odd function of the height difference with respect to the critical level, the reflection and transmission coefficients can be evaluated in closed form.

Over-reflection is possible for sufficiently small wavenumbers and Richardson numbers. It is pointed out that over-reflection and over-transmission cannot occur in a stable flow and that resonant over-reflection is not possible in our model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 120 , July 1982 , pp. 505 - 521
Copyright
© 1982 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Nat. Bureau of Standards.
Acheson, D. J. 1976 On over-reflexion. J. Fluid Mech. 77, 433–472.Google Scholar
Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear layer instability of an inviscid compressible fluid. Part 2. J. Fluid Mech. 71, 305–316.Google Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.Google Scholar
Brown, S. N. & Stewartson, K. 1980 On the non-linear reflexion of a gravity wave at a critical level. J. Fluid Mech. 100, 577–595.Google Scholar
Drazin, P. G. 1958 The stability of a shear layer in an unbounded heterogeneous inviscid fluid. J. Fluid Mech. 4, 214–224.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Drazin, P. G., Zaturska, M. B. & Banks, W. H. 1979 On the normal modes of parallel flow of inviscid stratified fluid. J. Fluid Mech. 95, 681–705.Google Scholar
Eltayeb, I. A. & McKenzie, J. F. 1975 Critical-level behaviour and wave amplification of a gravity wave incident upon a shear layer. J. Fluid Mech. 72, 661–671.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1953 Higher Transcendental Functions. McGraw-Hill.
Grimshaw, R. H. J. 1976 The reflection of internal gravity waves from a shear layer. Q. J. Mech. Appl. Math. 29, 511–525.Google Scholar
Grimshaw, R. H. J. 1979 On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile. J. Fluid Mech. 90, 161–178.Google Scholar
Heun, K. 1889 Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten. Math. Ann. 33, 161–179.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Howard, L. N. 1963 Neutral curves and stability boundaries in stratified flow. J. Fluid Mech. 16, 333–342.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.Google Scholar
Lighthill, M. J. 1960 Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil. Trans. R. Soc. Lond. A 252, 397–430.Google Scholar
Lindzen, R. S. 1974 Stability of a Helmholtz velocity profile in a continuously stratified, infinite Boussinesq fluid - applications to clear air turbulence. J. Atmos. Sci. 31, 1507–1514.Google Scholar
Mcintyre, M. E. & Weissman, M. A. 1978 On radiating instabilities and resonant overreflection. J. Atmos. Sci. 35, 1190–1196.Google Scholar
Mckenzie, J. F. 1972 Reflexion and amplification of acoustic gravity waves at a density and velocity discontinuity. J. Geophys. Res. 77, 2915–2926.Google Scholar
Mied, R. P. & Dugan, J. P. 1975 Internal wave reflection by a velocity shear and density anomaly. J. Phys. Oceanog. 5, 279–287.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.Google Scholar
Sluijter, F. W. 1967 Wave propagation through an Epstein profile across a static magnetic field. In Electromagnetic Wave Theory (ed. J. Brown), pp. 97–108. Pergamon.
Snow, C. 1952 Hypergeometric and Legendre functions with applications to integral equations of potential theory. Nat. Bureau of Standards. Appl. Math. Ser., vol. 19.
Thorpe, S. A. 1969 Neutral eigensolutions of the stability equation for stratified shear flow. J. Fluid Mech. 36, 673–683.Google Scholar
Van Duin, C. A. & Sluijter, F. W. 1979 An exact solution for the TM mode in an Eckart-Epstein type optical strip guide. In Antennas and Propagation, International Symposium Digest, vol. 1, pp. 304–305. I.E.E.E.
Van Duin, C. A. & Sluijter, F. W. 1980 Refractive index profiles with a local resonance based on Heun's equation. Radio Sci. 15, 95–103.Google Scholar