Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-17T22:54:25.696Z Has data issue: false hasContentIssue false
  • English
  • Français

Internal wave generation in uniformly stratified fluids. Part 2. Moving point sources

Published online by Cambridge University Press:  26 April 2006

Bruno Voisin
Bruno Voisin
Affiliation:
Laboratoire des Ecoulements Géophysiques et Industriels, Institut de Mécanique de Grenoble, CNRS – UJF – INPG, BP 53 X, 38041 Grenoble Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The Green's function method is applied to the generation of internal gravity waves by a moving point mass source. Arbitrary motion of a source of arbitrary time dependence is treated using the impulsive Green's function, while ‘classical’ approaches of uniform motion of a steady or oscillatory source are recovered using the monochromatic Green's function. Waves have locally the structure of impulsive waves, emitted at the retarded time tr, and having propagated with the group velocity; at each position and time an implicit equation defines tr, in terms of which the waves are written. A source both oscillating and moving generates two systems of waves, with respectively positive and negative frequencies, and when oscillations vanish these systems merge into one.

Three particular cases are considered: the uniform horizontal and vertical motions of a steady source, and the uniform horizontal motion of an oscillatory source. Waves spread downstream of the steady source. For the oscillatory source they can extend both upstream and downstream, depending on the ratio of the source frequency to the buoyancy frequency, and are contained inside conical wavefronts, parts of which are caustics. For horizontal motion, moreover, the steady analysis (based on the monochromatic Green's function) reveals the presence of two insignificant contributions overlooked by the unsteady analysis (based on the impulsive Green's function), but which for an extended source may become of the same order as the main contribution. Among those is an upstream columnar disturbance associated with blocking.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 261 , 25 February 1994 , pp. 333 - 374
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aksenov, A. V., Mozhaev, V. V., Skorovarov, V. E. & Sheronov, A. A. 1989 Stratified flow over a cylinder at low values of the internal Froude number. Fluid Dyn. 24, 639–642.Google Scholar
Baines, P. G. 1987 Upstream blocking and airflow over mountains. Ann. Rev. Fluid Mech. 19, 75–97.Google Scholar
Baines, P. G. & Grimshaw, R. H. J. 1979 Stratified flow over finite obstacles with weak stratification. Geophys. Astrophys. Fluid Dyn. 13, 317–334.Google Scholar
Baines, P. G. & Hoinka, K. P. 1985 Stratified flow over two-dimensional topography in fluid of infinite depth: a laboratory simulation. J. Atmos. Sci. 42, 1614–1630.Google Scholar
Belotserkovskii, O. M., Belotserkovskii, S. O., Gushchin, V. A., Morozov, E. N., Onufriev, A. T. & Ul'yanov, S. A. 1984 Numerical and experimental modeling of internal gravity waves during the motion of a body in a stratified liquid. Sov. Phys. Dokl. 29, 884–886.Google Scholar
Bleistein, N. 1966 Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Maths 19, 353–370.Google Scholar
Bleistein, N. 1984 Mathematical Methods for Wave Phenomena. Academic.
Blumen, W. & McGregor, C. D. 1976 Wave drag by three-dimensional mountain lee-waves in nonplanar shear flow. Tellus 28, 287–298.Google Scholar
Bonneton, P., Chomaz, J.-M. & Hopfinger, E. J. 1993 Internal waves produced by the turbulent wake of a sphere moving horizontally in a stratified fluid. J. Fluid Mech. 254, 23–40.Google Scholar
Boyer, D. L., Davies, P. A., Fernando, H. J. S. & Zhang, X. 1989 Linearly stratified flow past a horizontal circular cylinder. Phil. Trans. R. Soc. Lond. A 328, 501–528.Google Scholar
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
Brighton, P. W. M. 1978 Strongly stratified flow past three-dimensional obstacles. Q. J. R. Met. Soc. 104, 289–307.Google Scholar
Castro, I. P. 1987 A note on lee wave structures in stratified flow over three-dimensional obstacles. Tellus A 39, 72–81.Google Scholar
Castro, I. P. & Snyder, W. H. 1988 Upstream motions in stratified flow. J. Fluid Mech. 187, 487–506.Google Scholar
Castro, I. P., Snyder, W. H. & Marsh, G. L. 1983 Stratified flow over three-dimensional ridges. J. Fluid Mech. 135, 261–282.Google Scholar
Chashechkin, Yu, D. 1989 Hydrodynamics of a sphere in a stratified fluid. Fluid Dyn. 24, 1–7.Google Scholar
Chashechkin, Yu, D. & Makarov, S. A. 1984 Time-varying internal waves. Dokl. Earth Sci. Sect. 276, 210–213.Google Scholar
Cheng, H. K., Hefazi, H. & Brown, S. N. 1984 Topographically generated cyclonic disturbance and lee waves in a stratified rotating fluid. J. Fluid Mech. 141, 431–453.Google Scholar
Chomaz, J.-M., Bonneton, P. & Hopfinger, E. J. 1993 The structure of the near wake of a sphere moving horizontally in a stratified fluid. J. Fluid Mech. 254, 1–21.Google Scholar
Clark, T. L. & Peltier, W. R. 1977 On the evolution and stability of finite-amplitude mountain waves. J. Atmos. Sci. 34, 1715–1730.Google Scholar
Crapper, G. D. 1959 A three-dimensional solution for waves in the lee of mountains. J. Fluid. Mech. 6, 51–76.Google Scholar
Crapper, G. D. 1962 Waves in the lee of a mountain with elliptical contours. Phil. Trans. R. Soc. Lond. A 254, 601–623.Google Scholar
Crighton, D. G. & Oswell, J. E. 1991 Fluid loading with mean flow. I. Response of an elastic plate to localized excitation. Phil. Trans. R. Soc. Lond. A 335, 557–592.Google Scholar
Dokuchaev, V. P. & Dolina, I. S. 1977 Radiation of internal waves by sources in an exponentially stratified fluid. Izv. Atmos. Ocean. Phys. 13, 444–449.Google Scholar
Drazin, P. G. 1961 On the steady flow of a fluid of variable density past an obstacle. Tellus 13, 239–251.Google Scholar
Ekman, V. W. 1904 On dead water. In Norwegian North Polar Expedition, 1893-1896, Scientific Results (ed. F. Nansen), 5(15). Longmans (1906).
Foldvik, A. & Wurtele, M. G. 1967 The computation of the transient gravity wave. Geophys. J. R. Astron. Soc. 13, 167–185.Google Scholar
Gärtner, U. 1983a A note on the visualization and measurement of the internal wave field behind a cylinder moving through a stratified fluid. Geophys. Astrophys. Fluid Dyn. 26, 139–145.Google Scholar
Gärtner, U. 1983b Visualization of particle displacement and flow in stratified salt water. Exp. Fluids 1, 55–56.Google Scholar
Gärtner, U., Wernekinck, U. & Merzkirch, W. 1986 Velocity measurements in the field of an internal gravity wave by means of speckle photography. Exp. Fluids 4, 283–287.Google Scholar
Gilreath, H. E. & Brandt, A. 1985 Experiments on the generation of internal waves in a stratified fluid. AIAA J. 23, 693–700.Google Scholar
Gorodtsov, V. A. 1980 Radiation of internal waves during vertical motion of a body through a nonuniform liquid. J. Engng Phys. 39, 1062–1065.Google Scholar
Gorodtsov, V. A. 1981 Radiation of internal waves by rapidly moving sources in an exponentially stratified liquid. Sov. Phys. Dokl. 26, 229–230.Google Scholar
Gorodtsov, V. A. 1991 High-speed asymptotic form of the wave resistance of bodies in a uniformly stratified liquid. J. Appl. Mech. Tech. Phys. 32, 331–337.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. 1981 Two-dimensional problem for internal waves generated by moving singular sources. Fluid Dyn. 16, 219–224.Google Scholar
Gorodtsov, V. A. & Teodorovich, E. V. 1982 Study of internal waves in the case of rapid horizontal motion of cylinders and spheres. Fluid Dyn. 17, 893–898.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 Tables of Integrals, Series, and Products. 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, 398–402.Google Scholar
Grimshaw, R. H. J. 1969 Slow time-dependent motion of a hemisphere in a stratified fluid. Mathematika 16, 231–248.Google Scholar
Hanazaki, H. 1988 A numerical study of three-dimensional stratified flow past a sphere. J. Fluid Mech. 192, 393–419.Google Scholar
Hanazaki, H. 1989 Drag coefficient and upstream influence in three-dimensional stratified flow of finite depth. Fluid Dyn. Res. 4, 317–332.Google Scholar
Haussling, H. J. 1977 Viscous flows of stably stratified fluids over barriers. J. Atmos. Sci. 34, 589–602.Google Scholar
Hawthorne, W. R. & Martin, M. E. 1955 The effect of density gradient and shear on the flow over a hemisphere. Proc. R. Soc. Lond. A 232, 184–195.Google Scholar
Hefazi, H. T. & Cheng, H. K. 1988 The evolution of cyclonic disturbances and lee waves over a topography in a rapidly rotating stratified flow. J. Fluid Mech. 195, 57–76.Google Scholar
Hopfinger, E. J., Flor, J.-B., Chomaz, J.-M. & Bonneton, P. 1991 Internal waves generated by a moving sphere and its wake in a stratified fluid. Exp. Fluids 11, 255–261.Google Scholar
Hunt, J. C. R. & Snyder, W. H. 1980 Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech. 96, 671–704.Google Scholar
Janowitz, G. S. 1984 Lee waves in three-dimensional stratified flow. J. Fluid Mech. 148, 97–108.Google Scholar
Klemp, J. B. & Lilly, D. K. 1978 Numerical simulation of hydrostatic mountain waves. J. Atmos. Sci. 35, 78–107.Google Scholar
Landau, L. & Lifchitz, E. 1970 Théorie des Champs. Mir.
Lighthill, M. J. 1958 Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
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.
Lighthill, M. J. 1986 An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press.
Lin, Q., Lindberg, W. R., Boyer, D. L. & Fernando, H. J. S. 1992 Stratified flow past a sphere. J. Fluid Mech. 240, 315–354.Google Scholar
Makarov, S. A. & Chashechkin, Yu. D. 1981 Apparent internal waves in a fluid with exponential density distribution. J. Appl. Mech. Tech. Phys. 22, 772–779.Google Scholar
Makarov, S. A. & Chashechkin, Yu. D. 1982 Coupled internal waves in a viscous incompressible fluid. Izv. Atmos. Ocean. Phys. 18, 758–764.Google Scholar
Miles, J. W. 1969a The lee-wave régime for a slender body in a rotating flow. J. Fluid Mech. 36, 265–288.Google Scholar
Miles, J. W. 1969b Transient motion of a dipole in a rotating flow. J. Fluid Mech. 39, 433–442.Google Scholar
Miles, J. W. 1969c Waves and wave drag in stratified flow. In Proc. XIIth International Congress of Applied Mechanics (ed. M. Hétényi & W. G. Vincenti), pp. 50–76. Springer.
Miles, J. W. 1971 Internal waves generated by a horizontally moving source. Geophys. Fluid Dyn. 2, 63–87.Google Scholar
Miles, J. W. & Huppert, H. E. 1969 Lee waves in a stratified flow. Part 4. Perturbation approximations. J. Fluid Mech. 35, 497–525.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. Part I. McGraw-Hill.
Mowbray, D. E. & Rarity, B. S. H. 1967 The internal wave pattern produced by a sphere moving vertically in a density stratified liquid. J. Fluid Mech. 30, 489–495.Google Scholar
Murdock, J. W. 1977 The near-field disturbance created by a body in a stratified medium with a free surface. Trans. ASME E: J. Appl. Mech. 44, 534–540.Google Scholar
Peat, K. S. & Stevenson, T. N. 1975 Internal waves around a body moving in a compressible density-stratified fluid. J. Fluid Mech. 70, 673–688.Google Scholar
Peat, K. S. & Stevenson, T. N. 1976 The phase configuration of waves around a body moving in a rotating stratified fluid. J. Fluid Mech. 75, 647–656.Google Scholar
Peltier, W. R. & Clark, T. L. 1979 The evolution and stability of finite-amplitude mountain waves. Part II: Surface wave drag and severe downslope windstorms. J. Atmos. Sci. 36, 1498–1529.Google Scholar
Peltier, W. R. & Clark, T. L. 1983 Nonlinear mountain waves in two and three spatial dimensions. Q. J. R. Met. Soc. 109, 527–548.Google Scholar
Phillips, D. S. 1984 Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci. 41, 1073–1084.Google Scholar
Redekopp, L. G. 1975 Wave patterns generated by disturbances travelling horizontally in rotating stratified fluids. Geophys. Fluid Dyn. 6, 289–313.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
Rotunno, R. & Smolarkiewicz, P. K. 1991 Further results on lee vortices in low-Froude-number flow. J. Atmos. Sci. 48, 2204–2211.Google Scholar
Sarma, L. V. K. V. & Krishna, D. V. 1972 Motion of a sphere in a stratified fluid. Zastosow. Matem. 13, 123–130.Google Scholar
Scorer, R. S. 1956 Airflow over an isolated hill. Q. J. R. Met. Soc. 82, 75–81.Google Scholar
Sharman, R. D. & Wurtele, M. G. 1983 Ship waves and lee waves. J. Atmos. Sci. 40, 396–427.Google Scholar
Smith, R. B. 1980 Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus 32, 348–364.Google Scholar
Smith, R. B. 1988 Linear theory of stratified flow past an isolated mountain in isosteric coordinates. J. Atmos. Sci. 45, 3889–3896.Google Scholar
Smith, R. B. 1989 Mountain-induced stagnation points in hydrostatic flow. Tellus A 41, 270–274.Google Scholar
Smolarkiewicz, P. K. & Rotunno, R. 1989 Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci. 46, 1154–1164.Google Scholar
Stevenson, T. N. 1968 Some two-dimensional internal waves in a stratified fluid. J. Fluid Mech. 33, 715–720.Google Scholar
Stevenson, T. N. 1969 Axisymmetric internal waves generated by a travelling oscillating body. J. Fluid Mech. 35, 219–224.Google Scholar
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
Stevenson, T. N., Chang, W. L. & Laws, P. 1979 Viscous effects in lee waves. Geophys. Astrophys. Fluid Dyn. 13, 141–151.Google Scholar
Stevenson, T. N. & Thomas, N. H. 1969 Two-dimensional internal waves generated by a travelling oscillating cylinder. J. Fluid Mech. 36, 505–511.Google Scholar
Stevenson, T. N., Woodhead, T. J. & Kanellopulos, D. 1983 Viscous effects in some internal waves. Appl. Sci. Res. 40, 185–197.Google Scholar
Sturova, I. V. 1974 Wave motions produced in a stratified liquid from flow past a submerged body. J. Appl. Mech. Tech. Phys. 15, 796–805.Google Scholar
Sturova, I. V. 1978 Internal waves generated by local disturbances in a linearly stratified liquid of finite depth. J. Appl. Mech. Tech. Phys. 19, 330–336.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
Subba Rao, V. & Prabhakara Rao, G. V. 1971 On waves generated in rotating stratified liquids by travelling forcing effects. J. Fluid Mech. 46, 447–464.Google Scholar
Suzuki, M. & Kuwahara, K. 1992 Stratified flow past a bell-shaped hill. Fluid Dyn. Res. 9, 1–18.Google Scholar
Sykes, R. I. 1978 Stratification effects in boundary layer flow over hills. Proc. R. Soc. Lond. A 361, 225–243.Google Scholar
Sysoeva, E. Ya. & Chashechkin, Yu. D. 1991 Vortex systems in the stratified wake of a sphere. Fluid Dyn. 26, 544–551.Google Scholar
Thorpe, S. A. 1975 The excitation, dissipation, and interaction of internal waves in the deep ocean. J. Geophys. Res. 80, 328–338.Google Scholar
Trubnikov, B. N. 1959 The three-dimensional problem of the flow over a barrier of an air current unbounded at the top. Dokl. Earth Sci. Sect. 129, 1136–1138.Google Scholar
Umeki, M. & Kambe, T. 1989 Stream patterns of an isothermal atmosphere over an isolated mountain. Fluid Dyn. Res. 5, 91–109.Google Scholar
Vladimirov, V. A. & Il'in, K. I. 1991 Slow motions of a solid in a continuously stratified fluid. J. Appl. Mech. Tech. Phys. 32, 194–200.Google Scholar
Voisin, B. 1991a Rayonnement des ondes internes de gravité. Application aux corps en mouvement. Ph.D. thesis, Université Pierre et Marie Curie, Paris.
Voisin, B. 1991b Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources. J. Fluid Mech. 231, 439–480 (referred to herein as I).Google Scholar
Warren, F. W. G. 1960 Wave resistance to vertical motion in a stratified fluid. J. Fluid Mech. 7, 209–229.Google Scholar
Woodhead, T. J. 1983 The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid. Wave Motion 5, 157–165.Google Scholar
Wu, T. Y.-T. 1965 Three-dimensional internal gravity waves in a stratified free-surface flow. Z. Angew. Math. Mech. Sond. 45, T194–T195.Google Scholar
Wurtele, M. G. 1957 The three-dimensional lee wave. Beitr. Phys. Atmos. 29, 242–252.Google Scholar