Journal of Fluid Mechanics



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Nonlinear effects in reflecting and colliding internal wave beams


ALI TABAEI a1, T. R. AKYLAS a2 and KEVIN G. LAMB a3
a1 Department of Civil & Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
a2 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
a3 Department of Applied Mathematics, University of Waterloo, Ontario, Canada N2L 3G1

Article author query
tabaei a   [Google Scholar] 
akylas tr   [Google Scholar] 
lamb kg   [Google Scholar] 
 

Abstract

Using small-amplitude expansions, we discuss nonlinear effects in the reflection from a sloping wall of a time-harmonic (frequency $\omega$) plane-wave beam of finite cross-section in a uniformly stratified Boussinesq fluid with constant buoyancy frequency $N_{0}$. The linear solution features the incident and a reflected beam, also of frequency $\omega$, that is found on the same (opposite) side to the vertical as the incident beam if the angle of incidence relative to the horizontal is less (greater) than the wall inclination. As each of these beams is an exact nonlinear solution, nonlinear interactions are confined solely in the vicinity of the wall where the two beams meet. At higher orders, this interaction region acts as a source of a mean and higher-harmonic disturbances with frequencies $n\omega$ ($n\,{=}\,2,3,\ldots$); for $n\omega\,{<}\,N_{0}$ the latter radiate in the far field, forming additional reflected beams along $\sin^{-1}(n\omega/N_{0})$ to the horizontal. Depending on the flow geometry, higher-harmonic beams can be found on the opposite side of the vertical from the primary reflected beam. Using the same approach, we also discuss collisions of two beams propagating in different directions. Nonlinear interactions in the vicinity of the collision region induce secondary beams with frequencies equal to the sum and difference of those of the colliding beams. The predictions of the steady-state theory are illustrated by specific examples and compared against unsteady numerical simulations.

(Received February 26 2004)
(Revised October 5 2004)



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