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On the theory of nonlinear wave-wave interactions among geophysical waves

Published online by Cambridge University Press:  20 April 2006

P. Ripa
P. Ripa
Affiliation:
The Joint Institute for the Study of the Atmosphere and the Ocean, PMEL and UW, 3711 15th Ave NE, Seattle, Wa 98105 Present address: CICESE, ap. postal 2732, Ensenada, Baja California, Mexico.
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Abstract

The problem of nonlinear wave-wave interactions is reformulated, in a Eulerian framework, for two classical geophysical systems: barotropic Rossby waves and internal gravity waves on a vertical plane. The departure of the dynamical fields from the equilibrium state is expanded in the linear-problem eigenfunctions, using their properties of orthogonality and completeness. The system is then completely described by the expansion amplitudes, whose evolution is controlled by a system of equations (with quadratic nonlinearity) which is an exact representation of the original model equations. There is no a priori need for the usual multiple-time-scale analysis, or any other perturbation expansion, to develop the theory. These or other approximations (like truncation of the expansion basis or the Boltzmann equation for a stochastic description) can, if desired, be performed afterwards.

The evolution of the system is constrained mainly by the conservation of energy E and pseudo-momentum P, properties related to time and space homogeneity of the model equations. Conservation of E and P has, in turn, some interesting consequences: (a) a generalization of Fjortoft's theorem, (b) a class of exact nonlinear solutions (which includes the system of one single wave), and (c) conservation of E and P in an arbitrarily truncated system (which is useful in the development of approximations of the problem).

The properties of all possible resonant triads are shown and used to estimate the order of magnitude of off-resonant coupling coefficients.

The results are used in two different problems: the stability of a single wave (maximum growth rates are evaluated in both the strong- and weak-interactions limits) and the three-wave system. The general solution (for any initial condition and for both the resonant and off-resonant cases) of the latter is presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 103 , February 1981 , pp. 87 - 115
Copyright
© 1981 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1966 Handbook of Mathematical Functions. Dover.
Andrews, D. G. & Mcintyre, M. E. 1978 On wave-action and its relatives. J. Fluid Mech. 89, 647–664.Google Scholar
Bretherton, F. P. 1964 Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20, 457–479.Google Scholar
Craik, A. D. D. & Adam, J. A. 1979 ‘Explosive’ resonant wave interactions in a three-layer fluid flow. J. Fluid Mech. 92, 15–33.Google Scholar
Gelfand, I. M. & Fomin, S. V. 1963 Calculus of Variations. Prentice-Hall.
Gill, A. E. 1974 The stability of planetary waves on infinite beta-plane. Geophys. Fluid Dyn. 6, 29–47.Google Scholar
Hasselmann, K. 1967 Nonlinear interactions treated by the methods of theoretical physics (with application to the generation of waves by wind). Proc. Roy. Soc. A 299, 77–100.Google Scholar
Holloway, G. & Hendershott, M. C. 1977 Stochastic closure for nonlinear Rossby waves. J. Fluid Mech. 82, 747–765.Google Scholar
Holloway, G. 1980 Oceanic internal waves are not weak waves. J. Phys. Ocean. 10, 906–914.Google Scholar
Longuet-higgins, M. S. & Gill, A. E. 1967 Resonant interactions between planetary waves. Proc. Roy. Soc. A 299, 120–140.Google Scholar
Milder, M. 1976 A conservation law for internal gravity waves. J. Fluid Mech. 78, 209–216.Google Scholar
Mccomas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82, 1397–1412.Google Scholar
Mcgoldrick, L. F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305–331.Google Scholar
Orlanski, I. 1971 Energy spectrum of small-scale internal gravity waves. J. Geophys. Res. 76, 5829–5835.Google Scholar
Orlanski, I. 1973 Trapeze instability as a source of internal gravity waves. Part 1. J. Atmos. Sci. 30, 1007–1016.Google Scholar
Orlanski, I. & Cerasoli, C. P. 1980 Energy transfer among internal gravity modes: weak and strong interactions. J. Geophys. Res. (to appear).Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. J. Fluid Mech. 9, 193–217.Google Scholar
Ripa, P. 1980a Nonlinear wave—wave interactions in the equatorial beta-plane. (In preparation.)
Ripa, P. 1980b Conservation laws for ocean models in the beta-plane and the sphere. (In preparation.)