Resonant over-reflection of internal gravity waves from a thin shear layer
- 1 August 1981
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Fluid Mechanics
- Vol. 109, 349-365
- https://doi.org/10.1017/s0022112081001110
Abstract
In a previous paper (Grimshaw 1979) the resonant over-reflection of internal gravity waves from a vortex sheet was considered in the weakly nonlinear regime. It was shown there that the time evolution of the amplitude of the vortex sheet displacement was balanced by a cubic nonlinearity. For one vortex sheet mode, symmetrical with respect to the interface, it was shown that a steady finite-amplitude wave was possible. For the other, asymmetric modes, a singularity develops in a finite time. In the present paper, that analysis is extended by replacing the vortex sheet with a thin shear layer of thickness α2, where α is the amplitude of the shear layer displacement. The effect of this extension is to introduce a linear growth rate term in the amplitude equation, which is otherwise unaltered. The linear growth rate can be computed from a formula due to Drazin & Howard (1966, p. 67). The effect on the modes is that the symmetric mode is linearly damped and requires sustained forcing to be observed, while the asymmetric modes are slightly destabilized by the linear term and, as in the vortex-sheet model, develop a singularity in finite time.Keywords
This publication has 11 references indexed in Scilit:
- Modulation of an internal gravity wave packet in a stratified shear flowWave Motion, 1981
- The Concept of Wave Overreflection and Its Application to Baroclinic InstabilityJournal of the Atmospheric Sciences, 1980
- On the normal modes of parallel flow of inviscid stratified fluid. Part 2. Unbounded flow with propagation at infinityJournal of Fluid Mechanics, 1979
- On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profileJournal of Fluid Mechanics, 1979
- An exact theory of nonlinear waves on a Lagrangian-mean flowJournal of Fluid Mechanics, 1978
- On over-reflexionJournal of Fluid Mechanics, 1976
- On the Characteristics of Gravity Waves Generated by Atmospheric Shear LayersJournal of the Atmospheric Sciences, 1976
- Resonant Parallel Shear Instability in the Stably Stratified Planetary Boundary LayerJournal of the Atmospheric Sciences, 1976
- The Destabilizing Effect of the Ground on Kelvin-Helmholtz Waves in the AtmosphereJournal of the Atmospheric Sciences, 1976
- Hydrodynamic Stability of Parallel Flow of Inviscid FluidPublished by Elsevier ,1966