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Resonant interactions between topographic planetary waves in a continuously stratified fluid

Published online by Cambridge University Press:  12 April 2006

Lawrence A. Mysak
Lawrence A. Mysak
Affiliation:
National Center for Atmospheric Research, Boulder, Colorado 80307
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Abstract

The resonant interactions between topographic planetary waves in a continuously stratified fluid are investigated theoretically. The interacting waves form a resonant triad and travel along a channel with a uniformly sloping bottom. The basic state stratification in the channel is characterized by a constant buoyancy frequency. The existence of solutions to the quadratic resonance conditions is established graphically. Each wave by itself is a bottom-intensified oscillation of the type discovered by Rhines (1970) except for the addition of a small positive frequency correction. This correction must be included to satisfy higher-order terms in the bottom boundary condition. For strong stratification (r2 [Gt ] L2, where r = internal deformation radius and L = channel width), the waves are strongly bottom-trapped and this frequency correction is negligible. For weak stratification (r2 [Lt ] L2) the waves are barotropic and the frequency correction is O(δ), where δ = fractional change in depth across the channel. In many oceanic contexts, δ lies in the range 0·1-0·4 and therefore this correction can produce a significant change in the phase speed. The amplitudes of the waves in the triad obey the classical gyroscopic equations usually encountered in quadratic resonance problems. In particular, the amplitudes evolve on the slow time scale \[ t=O(1/f_0\delta^2), \] which for our scaling assumptions is also O(1/f0Ro), where Ro is the Rossby number.

The results are applied to the Norwegian continental slope region. It is shown that, in this vicinity, there may exist resonant triads consisting of two short, high-frequency waves (periods around 3-4 days) and one long, low-frequency wave (period around 9 days).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 84 , Issue 4 , 27 February 1978 , pp. 769 - 793
Copyright
© 1978 Cambridge University Press

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