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Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources

Published online by Cambridge University Press:  26 April 2006

Bruno Voisin
Bruno Voisin
Affiliation:
Thomson-Sintra Activatesés Sous-Marines, i, avenue A. Briand, 94117 Arcueil, France and Département de Recherches Physiques, Université Pierre et Marie Curie. 4, place Jussieu, 75252 Paris, France
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Abstract

In both Boussinesq and non-Boussinesq cases the Green's function of internal gravity waves is calculated, exactly for monochromatic waves and asymptotically for impulsive waves. From its differentiation the pressure and velocity fields generated by a point source are deduced. by the same method the Boussinesq monochromatic and impulsive waves radiated by a pulsating sphere are investigated.

Boussinesq monochromatic waves of frequency ω < N are confined between characteristic cones θ = arccos(ω/N) tangent to the source region (N being the buoyancy frequency and θ the observation angle from the vertical). In that zone the point source model is inadequate. For the sphere an explicit form is given for the waves, which describes their conical 1/r½ radial decay and their transverse phase variations.

Impulsive waves comprise gravity and buoyancy waves, whose separation process is non-Boussinesq and follows the arrival of an Airy wave. As time t elapses, inside the torus of vertical axis and horizontal radius 2Nt/β for gravity waves and inside the circumscribing cylinder for buoyancy waves, both components become Boussinesq and have wavelengths negligible compared with the scale height 2/β of the stratification. Then, gravity waves are plane propagating waves of frequency N cos θ, and buoyancy waves are radial oscillations of the fluid at frequency N; for the latter, initially propagating waves comparable with gravity waves, the horizontal phase variations have vanished and the amplitude has become insignificant as the Boussinesq zone has been entered. In this zone, outside the torus of vertical axis and horizontal radius Nta, a sphere of radius a [Lt ] 2/β is compact compared with the wavelength of the dominant gravity waves. Inside the torus gravity waves vanish by destructive interference. For the remaining buoyancy oscillations the sphere is compact outside the vertical cylinder circumscribing it, whereas the fluid is quiescent inside this cylinder.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 231 , October 1991 , pp. 439 - 480
Copyright
© 1991 Cambridge University Press

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References

Abramowitz, M. & Stegun, T. A. (eds.) 1965 Handbook of Mathematical Functions. Dover.
Appleby, J. C. & Crighton, D. G. 1986 Non-Boussinesq effects in the diffraction of internal waves from an oscillating cylinder. Q. J. Mech. Appl. Maths 39, 209–231.Google Scholar
Appleby, J. C. & Crighton, D. G. 1987 Internal gravity waves generated oscillations of a sphere. J. Fluid Mech. 183, 439–450.Google Scholar
Baines, P. G. 1971 The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid Mech. 46, 273–291.Google Scholar
Barcilon, V. & Bleistein, N. 1969 Scattering of inertial waves in a rotating fluid. Stud. Appl. Maths 48, 91–104.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bleistein, N. 1984 Mathematical Methods for Wave Phenomena. Academic.
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals. Dover.
Brekhovskikh, L. & Goncharov, V. 1985 Mechanics of Continua and Wave Dynamics. Springer.
Bretherton, F. P. 1967 The time-dependent motion due to a cylinder moving in an unbounded rotating or stratified fluid. J. Fluid Mech. 28, 545–570.Google Scholar
Chashechkin, Yu. D. & Makarov, S. A. 1984 Time-varying internal waves. Dokl. Earth Sci. Sect. 276, 210–213.Google Scholar
Cole, J. D. & Greifinger, C. 1969 Acoustic-gravity waves from an energy source at the ground in an isothermal atmosphere. J. Geophys. Res. 74, 3693–3703.Google Scholar
Dickinson, R. E. 1969 Propagators of atmospheric motions. 1. Excitation by point impulses. Rev. Geophys. 7, 483–514.Google Scholar
Felsen, L. B. 1969 Transients in dispersive media, Part I: Theory. IEEE Trans. Antennas Propag. 17, 191–200.Google Scholar
Gordon, D. & Stevenson, T. N. 1972 Viscous effects in a vertically propagating internal wave. J. Fluid Mech. 56, 629–639.Google Scholar
Gorodtsov, V. A. & Teodorovich, E. V. 1980 On the generation of internal waves in the presence of uniform straight-line motion of local and nonlocal sources. Izv. Atmos, Ocean. Phys. 16, 699–704.Google Scholar
Gorodtsov, V. A. & Teodorovich, E. V. 1983 Radiation of internal waves by periodically moving sources. J. Appl. Mech. Tech. Phys. 24, 521–526.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals. Series, and Products (Corrected and Enlarged Edition). Academic.
Grigor'ev, G. I. & Dokuchaev, V. P. 1970 On the theory of the radiation of acoustic-gravity waves by mass sources in a stratified isothermal atmosphere. Izv. Atmos. Ocean. Phys. 6, 398–102.Google Scholar
Grimshaw, R. 1969 Slow time-dependent motion of a hemisphere in a stratified fluid. Mathematika 16, 231–248.Google Scholar
Hart, R. W. 1981 Generalized scalar potentials for linearized three-dimensional flows with vorticity. Phys. Fluids 24, 1418–1420.Google Scholar
Hendershott, M. C. 1969 Impulsively started oscillations in a rotating stratified fluid. J. Fluid Mech. 36, 513–527.Google Scholar
Hurley, D. G. 1972 A general method for solving steady-state internal gravity wave problems. J. Fluid Mech. 56, 721–740.Google Scholar
Jackson, J. D. 1975 Classical Electrodynamics(2nd edn). Wiley.
Kamachi, M. & Honji, H. 1988 Interaction of interfacial and internal waves, Fluid Dyn. Res. 2, 229–241.Google Scholar
Kato, S. 1966a The response of an unbounded atmosphere to point disturbances. I. Time-harmonic disturbances. Astrophys. J. 143, 893–903.Google Scholar
Kato, S. 1966b The response of an unbounded atmosphere to point disturbances. II. Impulsive disturbances. Astrophys, J. 144, 326–336.Google Scholar
Krishna, D. V. & Sarma, L. V. K. V. 1969 Motion of an axisymmetric body in a rotating stratified fluid confined between two parallel planes. J. Fluid Mech. 38, 833–842.Google Scholar
Larsen, L. H. 1969 Oscillations of a neutrally buoyant sphere in a stratified fluid. Deep Sea Res. 16, 587–603.Google Scholar
Lighthill, M. J. 1958 Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Lighthill, M. J. 1960 Studies on magneto-hydrodynamics waves and other anisotropic wave motions. Phil. Trans. R. Soc. Lond. A 252, 397–430.Google Scholar
Lighthill, M. J. 1965 Group velocity. J. Inst. Maths Applies 1, 1–28.Google Scholar
Lighthill, M. J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27, 725–752.Google Scholar
Lighthill, M. J. 1978 Waves in Fluids, Cambridge University Press.
Liu, C. H. & Yeh, K. C. 1971 Excitation of acoustic-gravity waves in an isothermal atmosphere. Tellus 23, 150–163.Google Scholar
McLaren, T. I., Pierce, A. D., Fohl, T. & Murphy, B. L. 1973 An investigation of internal gravity waves generated by a buoyantly rising fluid in a stratified medium. J. Fluid Mech. 57, 229–240.Google Scholar
Miles, J. W. 1971 Internal waves generated by a horizontally moving source Geophys. Fluid Dyn. 2, 63–87.Google Scholar
Miropol'skii, Yu. Z. 1978 Self-similar solutions of the Cauchy problem for internal waves in an unbounded fluid. Izv. Atmos. Ocean. Phys. 14, 673–679.Google Scholar
Moore, D. W. & Spiegel, E. A. 1964 The generation and propagation of waves in a compressible atmosphere. Astrophys. J. 139, 48–71.Google Scholar
Morgan, G. W. 1953 Remarks on the problem of slow motions in a rotating fluid. Proc. Camb. Phil. Soc. 49, 362–364.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. Part I. McGraw-Hill.Google Scholar
Mowbray, D. E. & Rarity, B. S. H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16.Google Scholar
Pierce, A. D. 1963 Propagation of acoustic-gravity waves from a small source above the ground in an isothermal atmosphere. J. Acoust. Soc. Am. 35, 1798–1807.Google Scholar
Pierce, A. D. 1981 Acoustics. An Introduction to its Physical Principles and Applications. McGraw-Hill.
Ramachandra Rao, A. 1973 A note on the application of a radiation condition for a source in a rotating stratified fluid. J. Fluid Mech. 58, 161–164.Google Scholar
Ramachandra Rao, A. 1975 On an oscillatory point force in a rotating stratified fluid. J. Fluid Mech. 72, 353–362.Google Scholar
Rehm, R. G. & Radt, H. S. 1975 Internal waves generated by a translating oscillating body. J. Fluid Mech. 68, 235–258.Google Scholar
Row, R. V. 1967 Acoustic-gravity- waves in the upper atmosphere due to a nuclear detonation and an earthquake. J. Geophys. Res. 72, 1599–1610.Google Scholar
Sarma, L. V. K. V. & Krishna, D. V. 1972 Oscillation of axisymmetric bodies in a stratified fluid. Zastosow. Matem. 13, 109–121.Google Scholar
Sarma, L. V. K. V. & Naidu, K. B. 1972a Source in a rotating stratified fluid. Acta Mech. 13, 21–29.Google Scholar
Sarma, L. V. K. V. & Naidu, K. B. 1972 Closed form solution for a point force in a rotating stratified fluid. Pure Appl. Geophys. 99, 156–168.Google Scholar
Sekerzh-Zen'kovich, S. Y. A. 1979 A fundamental solution of the internal-wave operator. Sov. Phys. Dokl. 24, 347–349.Google Scholar
Sekerzh-Zen'kovich, S. Ya. 1981 Construction of the fundamental solution for the operator of internal waves. Appl. Maths Mech. 45, 192–198.Google Scholar
Sekerzh-zen'kovich, S. Ya. 1982 Cauchy problem for equations of internal waves. Appl. Maths Mech. 46, 758–764.Google Scholar
Sobolev, S. L. 1965 Sur une classe des problèmes de physique mathématique. In Simposio Internazionale Sulle A pplicazioni Dell' Analisi Alla Fisica Matematica. Cagliari—Sassari, September 28—October 4 1964, Edizioni Cremonese (Roma), pp. 192–208.
Stevenson, T. N. 1973 The phase configuration of internal waves around a body moving in a density stratified fluid. J. Fluid Mech. 60, 759–767.Google Scholar
Stewartson, K. 1952 On the slow motion of a sphere along the axis of a rotating fluid. Proc. Camb. Phil. Soc. 48, 168–177.Google Scholar
Sturova, I. V. 1980 Internal waves generated in an exponentially stratified fluid by an arbitrarily moving source. Fluid Dyn. 15, 378–383.Google Scholar
Teodorovich, E. V. & Gorodtsov, V. A. 1980 On some singular solutions of internal wave equations. Izv. Atmos. Ocean. Phys. 16, 551–553.Google Scholar
Tolstoy, I. 1973 Wave Propagation. McGraw-Hill.
Voisin, B. 1991 Rayonnement des ondes internes de gravité. Application aux corps en mouvement. Ph.D. thesis, Université Pierre et Marie Curie.
Watson, G. N. 1966 A Treatise on the Theory of Bessel Functions (2nd edn). Cambridge University Press.
Zavol'skii, N. A. & Zaitsev, A. A. 1984 Development of internal waves generated by a concentrated pulse source in an infinite uniformly stratified fluid. J. Appl. Mech. Tech. Phys. 25, 862–867.Google Scholar