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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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