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Lee waves from a sphere in a stratified flow

Published online by Cambridge University Press:  15 February 2007

B. VOISIN
B. VOISIN*
Affiliation:
Laboratoire des Écoulements Géophysiques et Industriels, UJF, INPG, CNRS; BP 53, 38041 Grenoble, FranceBruno.Voisin@hmg.inpg.fr
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Abstract

Two asymptotic analyses of the generation of lee waves by horizontal flow at velocity U of a stratified fluid of buoyancy frequency N past a sphere of radius a are presented, for either weak or strong stratification, corresponding to either large or small internal Froude number F=U/(Na), respectively. For F⋙1, the fluid separates into two regions radially: an inner region of scale a with three-dimensional irrotational flow unaffected by the stratification, and an outer region of scale U/N with small-amplitude lee waves generated by the O(1) vertical motion in the inner region. For F⋘1, the fluid separates into five layers vertically: from the lower dividing streamsurface situated at a distance U/N above the bottom of the sphere to the upper dividing streamsurface situated at a distance U/N below the top, there is a middle layer with two-dimensional horizontal irrotational flow; from the upper dividing streamsurface to the top of the sphere, and from the lower dividing streamsurface to the bottom, there are top and bottom transition layers, respectively, with three-dimensional flow; above the top and below the bottom, there are upper and lower layers, respectively, with small-amplitude lee waves generated by the O(F) vertical motion in the transition layers.

The waves are calculated where they have small amplitudes. The forcing is represented by a source of mass: for F⋙1, the surface distribution of singularities equivalent to the sphere in three-dimensional irrotational flow; for F⋘1, the horizontal distribution of singularities equivalent, in the upper (resp. lower) layer, to the flat cut-off obstacle made up of the top (resp. bottom) portion of the sphere protruding above (resp. below) the upper (resp. lower) dividing streamsurface. The analysis is validated by comparison of the theoretical wave drag with existing experimental determinations. For F⋙1, the drag coefficient decreases as (ln F+7/4-γ)/(4F4), with γ the Euler constant; for F⋘1, it increases as . The waves have the crescent shape of the three-dimensional lee waves from a dipole, modulated by interferences associated with the finite size of the forcing. For strong stratification, the hydrostatic approximation is seen to produce correct leading-order drag, but incorrect waves.

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Journal of Fluid Mechanics , Volume 574 , 10 March 2007 , pp. 273 - 315
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Copyright © Cambridge University Press 2007

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References

REFERENCES

Aksenov, A. V., Gorodtsov, V. A. & Sturova, I. V. 1986 Modeling of the flow of a stratified ideal incompressible fluid around a cylinder. Preprint No. 282, Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow (in Russian).Google Scholar
Akylas, T. R. & Davis, K. S. 2001 Three-dimensional aspects of nonlinear stratified flow over topography near the hydrostatic limit. J. Fluid Mech. 428, 81–105.CrossRefGoogle Scholar
Baines, P. G. 1987 Upstream blocking and airflow over mountains. Annu. Rev. Fluid Mech. 19, 75–97.CrossRefGoogle Scholar
Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.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.CrossRefGoogle Scholar
Baines, P. G. & Smith, R. B. 1993 Upstream stagnation points in stratified flow past obstacles. Dyn. Atmos. Ocean. 18, 105–113.CrossRefGoogle Scholar
Barnes, E. W. 1906 The asymptotic expansion of integral functions defined by Taylor's series. Phil. Trans. R. Soc. Lond. A 206, 249–297.Google Scholar
Barnes, E. W. 1907 The asymptotic expansion of integral functions defined by generalised hypergeometric series. Proc. Lond. Math. Soc. (2) 5, 59–116.CrossRefGoogle Scholar
Barnes, E. W. 1908 On functions defined by simple types of hypergeometric series. Trans. Camb. Phil. Soc. 20, 253–279.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals, 2nd edn. Dover.Google Scholar
Blumen, W. 1965 A random model of momentum flux by mountain waves. Geofys. Publ. 26 (2), 1–33.Google Scholar
Blumen, W. & McGregor, C. D. 1976 Wave drag by three-dimensional mountain lee-waves in nonplanar shear flow. Tellu. 28, 287–298.CrossRefGoogle 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.CrossRefGoogle Scholar
Bonneton, P., Chomaz, J.-M., Hopfinger, E. J. & Perrier, M. 1996 The structure of the turbulent wake and the random internal wave field generated by a moving sphere in a stratified fluid. Dyn. Atmos. Ocean. 23, 299–308.CrossRefGoogle 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.CrossRefGoogle Scholar
Brighton, P. W. M. 1978 Strongly stratified flow past three-dimensional obstacles. Q. J. R. Met. Soc. 104, 289–307.Google Scholar
Broutman, D. & Rottman, J. W. 2004 A simplified Fourier method for computing the internal wavefield generated by an oscillating source in a horizontally moving, depth-dependent background. Phys. Fluid. 16, 3682–3689.CrossRefGoogle Scholar
Castro, I. P. 1987 A note on lee wave structures in stratified flow over three-dimensional obstacles. Tellus A 39, 72–81.CrossRefGoogle Scholar
Castro, I. P., Snyder, W. H. & Marsh, G. L. 1983 Stratified flow over three-dimensional ridges. J. Fluid Mech. 135, 261–282.CrossRefGoogle Scholar
Castro, I., Vosper, S., Paisley, M. & Hayden, P. 2001 Vortex shedding behind tapered obstacles in neutral and stratified flow. Dyn. Atmos. Ocean. 34, 145–163.CrossRefGoogle Scholar
Chashechkin, Yu. D. 1989 Hydrodynamics of a sphere in a stratified fluid. Fluid Dyn. 24, 1–7. [Transl. from Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (1), 3–9.]CrossRefGoogle Scholar
Chomaz, J.-M., Bonneton, P., Butet, A., Perrier, M. & Hopfinger, E. J. 1992 Froude number dependence of the flow separation line on a sphere towed in a stratified fluid. Phys. Fluids A 4, 254–258.CrossRefGoogle 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.CrossRefGoogle Scholar
Crapper, G. D. 1959 A three-dimensional solution for waves in the lee of mountains. J. Fluid Mech. 6, 51–76.CrossRefGoogle 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
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. [Transl. from Izv. Akad. Nauk SSSR Fiz. Atmos. Okean. 13, 655–663.]Google Scholar
Drazin, P. G. 1961 On the steady flow of a fluid of variable density past an obstacle. Tellu. 13, 239–251.CrossRefGoogle Scholar
Drazin, P. G. & Su, C. H. 1975 A note on long-wave theory of airflow over a mountain. J. Atmos. Sci. 32, 437–439.2.0.CO;2>CrossRefGoogle Scholar
Dupont, P., Kadri, Y. & Chomaz, J.-M. 2001 Internal waves generated by the wake of Gaussian hills. Phys. Fluid. 13, 3223–3233.CrossRefGoogle Scholar
Dupont, P. & Voisin, B. 1996 Internal waves generated by a translating and oscillating sphere. Dyn. Atmos. Ocean. 23, 289–298.CrossRefGoogle Scholar
Eames, I. & Hunt, J. C. R. 1997 Inviscid flow around bodies moving in weak density gradients without buoyancy effects. J. Fluid Mech. 353, 331–355.CrossRefGoogle Scholar
Egan, B. A. 1984 Transport and diffusion in complex terrain (review). Boundary-Layer Met. 30, 3–28.CrossRefGoogle Scholar
Gilreath, H. E. & Brandt, A. 1985 Experiments on the generation of internal waves in a stratified fluid. AIAA J. 23, 693–700.CrossRefGoogle 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. [Transl. from Inzh.-Fiz. Zh. 39, 619–623.]CrossRefGoogle Scholar
Gorodtsov, V. A., Reznik, S. N. & Stepanyants, Yu. A. 1997 Radiative forces acting on point sources moving in a stratified fluid. J. Expl Theor. Phys. 85, 276–284. [Transl. from Zh. Éksp. Teor. Fiz. 112, 507–523.]CrossRefGoogle Scholar
Gorodtsov, V. A. & Teodorovich, É. 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. [Transl. from Izv. Akad. Nauk SSSR Fiz. Atmos. Okean. 16, 954–961.]Google Scholar
Gorodtsov, V. A. & Teodorovich, É. V. 1981 Two-dimensional problem for internal waves generated by moving singular sources. Fluid Dyn. 16, 219–224. [Transl. from Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (2), 77–83.]CrossRefGoogle Scholar
Gorodtsov, V. A. & Teodorovich, É. V. 1982 Study of internal waves in the case of rapid horizontal motion of cylinders and spheres. Fluid Dyn. 17, 893–898. [Transl. from Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (6), 94–100.]CrossRefGoogle Scholar
Gorodtsov, V. A. & Teodorovich, É. V. 1983 Radiation of internal waves by periodically moving sources. J. Appl. Mech. Tech. Phys. 24, 521–526. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (4), 81–87.]Google Scholar
Gourlay, M. J., Arendt, S. C., Fritts, D. C. & Werne, J. 2001 Numerical modeling of initially turbulent wakes with net momentum. Phys. Fluid. 13, 3783–3802.CrossRefGoogle Scholar
Greenslade, M. D. 1992 Strongly stratified airflow over and around mountains. Ph.D. thesis, University of Leeds.Google Scholar
Greenslade, M. D. 1994 Strongly stratified airflow over and around mountains. In Stably Stratified Flows: Flow and Dispersion over Topography (ed. Castro, I. P. & Rockliff, N. J.), pp. 25–37. Oxford University Press.Google Scholar
Greenslade, M. D. 2000 Drag on a sphere moving horizontally in a stratified fluid. J. Fluid Mech. 418, 339–350.CrossRefGoogle Scholar
Grimshaw, R. 1969 Slow time-dependent motion of a hemisphere in a stratified fluid. Mathematik. 16, 231–248.CrossRefGoogle Scholar
Hanazaki, H. 1988 A numerical study of three-dimensional stratified flow past a sphere. J. Fluid Mech. 192, 393–419.CrossRefGoogle 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
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Hopfinger, E. J., Flór, J.-B., Chomaz, J.-M. & Bonneton, P. 1991 Internal waves generated by a moving sphere and its wake in a stratified fluid. Exps. Fluid. 11, 255–261.CrossRefGoogle Scholar
Hunt, J. C. R., Feng, Y., Linden, P. F., Greenslade, M. D. & Mobbs, S. D. 1997 Low-Froude-number stable flows past mountains. Nuovo Cim. C 20, 261–272.Google Scholar
Hunt, J. C. R. & Richards, K. J. 1984 Stratified airflow over one or two hills. Boundary-Layer Met. 30, 223–259.CrossRefGoogle Scholar
Hunt, J. C. R., Richards, K. J. & Brighton, P. W. M. 1988 Stably stratified shear flow over low hills. Q. J. R. Met. Soc. 114, 859–886.CrossRefGoogle 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.CrossRefGoogle Scholar
Huppert, H. E. & Miles, J. W. 1969 Lee waves in a stratified flow. Part 3. Semi-elliptical obstacle. J. Fluid Mech. 35, 481–496.CrossRefGoogle Scholar
Jackson, J. D. 1999 Classical Electrodynamics, 3rd edn. Wiley.Google Scholar
Janowitz, G. S. 1973 Unbounded stratified flow over a vertical barrier. J. Fluid Mech. 58, 375–388.CrossRefGoogle Scholar
Janowitz, G. S. 1984 Lee waves in three-dimensional stratified flow. J. Fluid Mech. 148, 97–108.CrossRefGoogle Scholar
Kadri, Y., Bonneton, P., Chomaz, J.-M. & Perrier, M. 1996 Stratified flow over three-dimensional topography. Dyn. Atmos. Ocean. 23, 321–334.CrossRefGoogle Scholar
Kantzios, Y. D. & Akylas, T. R. 1993 An asymptotic theory of nonlinear stratified flow of large depth over topography. Proc. R. Soc. Lond. A 440, 639–653.Google Scholar
Kozhevnikov, V. N. 1963 A single nonlinear problem of the orographic disturbance of stratified air flow. Bull. (Izv.) Acad. Sci. USSR Geophys. Ser., 675–680. [Transl. from Izv. Akad. Nauk SSSR Ser. Geofiz., 1108–1116.]Google Scholar
Kozhevnikov, V. N. 1968 Orographic perturbations in the two-dimensional stationary problem. Izv. Atmos. Ocean. Phys. 4, 16–27. [Transl. from Izv. Akad. Nauk SSSR Fiz. Atmos. Okean. 4, 33–52.]Google Scholar
Krishna, D. V. 1968 Unsteady stratified flow past a cylinder. Zastosow. Mat. 9, 417–427.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Butterworth–Heinemann.Google Scholar
Landau, L. D., Lifshitz, E. M. & Pitaevskii, L. P. 1984 Electrodynamics of Continuous Media, 2nd edn. Butterworth–Heinemann.Google Scholar
Laprise, R. & Peltier, W. R. 1989 On the structural characteristics of steady finite-amplitude mountain waves over bell-shaped topography. J. Atmos. Sci. 46, 586–595.2.0.CO;2>CrossRefGoogle 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.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lighthill, J. 1986 An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press.Google Scholar
Lilly, D. K. & Klemp, J. B. 1979 The effects of terrain shape on nonlinear hydrostatic mountain waves. J. Fluid Mech. 95, 241–261.CrossRefGoogle Scholar
Lin, Q., Boyer, D. L. & Fernando, H. J. S. 1993 Internal waves generated by the turbulent wake of a sphere. Exps. Fluid. 15, 147–154.CrossRefGoogle Scholar
Lin, Q., Lindberg, W. R., Boyer, D. L. & Fernando, H. J. S. 1992 Stratified flow past a sphere. J. Fluid Mech. 240, 315–354.CrossRefGoogle Scholar
Lofquist, K. E. B. & Purtell, L. P. 1984 Drag on a sphere moving horizontally through a stratified liquid. J. Fluid Mech. 148, 271–284.CrossRefGoogle Scholar
Long, R. R. 1953 Some aspects of the flow of stratified fluids. I. A theoretical investigation. Tellu. 5, 42–58.CrossRefGoogle Scholar
MacKinnon, R. F., Mulley, R. & Warren, F. W. G. 1969 Some calculations of gravity wave resistance in an inviscid stratified fluid. J. Fluid Mech. 38, 61–73.CrossRefGoogle 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. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (6), 47–54.]Google Scholar
Makarov, S. A. & Chashechkin, Yu. D. 1982 Coupled internal waves in a viscous incompressible fluid. Izv. Atmos. Ocean. Phys. 18, 758–764. [Transl. from Izv. Akad. Nauk SSSR Fiz. Atmos. Okean. 18, 986–994.]Google Scholar
Mason, P. J. 1977 Forces on spheres moving horizontally in a rotating stratified fluid. Geophys. Astrophys. Fluid Dyn. 8, 137–154.CrossRefGoogle Scholar
Mellin, R. H. 1910 Abriß einer einheitlichen Theorie der Gamma- und der hypergeometrischen Funktionen. Math. Annl. 68, 305–337.CrossRefGoogle Scholar
Miles, J. W. 1968 Lee waves in a stratified flow. Part 1. Thin barrier. J. Fluid Mech. 32, 549–567.CrossRefGoogle Scholar
Miles, J. W. 1969 Waves and wave drag in stratified flows. In Proc. XIIth Intl Congress of Applied Mechanics (ed. Hétényi, M. & Vincenti, W. G.), pp. 50–76. Springer.Google Scholar
Miles, J. W. 1971 Internal waves generated by a horizontally moving source. Geophys. Fluid Dyn. 2, 63–87.CrossRefGoogle Scholar
Miles, J. W. & Huppert, H. E. 1968 Lee waves in a stratified flow. Part 2. Semi-circular obstacle. J. Fluid Mech. 33, 803–814.CrossRefGoogle Scholar
Miles, J. W. & Huppert, H. E. 1969 Lee waves in a stratified flow. Part 4. Perturbation approximations. J. Fluid Mech. 35, 497–525.CrossRefGoogle Scholar
Miranda, P. M. A. & James, I. N. 1992 Non-linear three-dimensional effects on gravity-wave drag: Splitting flow and breaking waves. Q. J. R. Met. Soc. 118, 1057–1081.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.CrossRefGoogle Scholar
Newley, T. M. J., Pearson, H. J. & Hunt, J. C. R. 1991 Stably stratified rotating flow through a group of obstacles. Geophys. Astrophys. Fluid Dyn. 58, 147–171.CrossRefGoogle Scholar
Palierne, J. F. 1999 On the motion of rigid bodies in incompressible inviscid fluids of inhomogeneous density. J. Fluid Mech. 393, 89–98.CrossRefGoogle Scholar
Paris, R. B. & Kaminski, D. 2001 Asymptotics and Mellin–Barnes Integrals. Cambridge University Press.CrossRefGoogle 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.CrossRefGoogle Scholar
Phillips, D. S. 1984 Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci. 41, 1073–1084.2.0.CO;2>CrossRefGoogle Scholar
Prasad, D., Ramirez, J. & Akylas, T. R. 1996 Stability of stratified flow of large depth over finite-amplitude topography. J. Fluid Mech. 320, 369–394.CrossRefGoogle Scholar
Rottman, J. W., Broutman, D., Spedding, G. & Meunier, P. 2004 The internal wave field generated by the body and wake of a horizontally moving sphere in a stratified fluid. In Proc. 15th Australasian Fluid Mechanics Conf. (ed. Behnia, M., Lin, W. & McBain, G. D.), pap. AFMC00131. University of Sydney. Available in electronic form at the URL http://www.aeromech.usyd.edu.au/15afmc/proceedings/.Google Scholar
Rotunno, R., Grubišić, V. & Smolarkiewicz, P. K. 1999 Vorticity and potential vorticity in mountain wakes. J. Atmos. Sci. 56, 2796–2810.2.0.CO;2>CrossRefGoogle Scholar
Rotunno, R. & Smolarkiewicz, P. K. 1991 Further results on lee vortices in low-Froude number flow. J. Atmos. Sci. 48, 2204–2211.2.0.CO;2>CrossRefGoogle Scholar
Rowe, R. D., Benjamin, S. F., Chung, K. P., Havlena, J. J. & Lee, C. Z. 1981 Field studies of stable air flow over and around a ridge. Atmos. Environ. 16, 643–653.CrossRefGoogle Scholar
Ryan, W. & Lamb, B. 1984 Determination of dividing streamline heights and Froude numbers for predicting plume transport in complex terrain. J. Air Pollut. Control Ass. 31, 152–155.CrossRefGoogle Scholar
Sarma, L. V. K. V. & Krishna, D. V. 1972 Motion of a sphere in a stratified fluid. Zastosow. Mat. 13, 123–130.Google Scholar
Scase, M. M. & Dalziel, S. B. 2004 Internal wave fields and drag generated by a translating body in a stratified fluid. J. Fluid Mech. 498, 289–313.CrossRefGoogle Scholar
Schär, C. & Durran, D. R. 1997 Vortex formation and vortex shedding in continuously stratified flows past isolated topography. J. Atmos. Sci. 54, 534–554.2.0.CO;2>CrossRefGoogle Scholar
Scorer, R. S. 1956 Airflow over an isolated hill. Q. J. R. Met. Soc. 82, 75–81.CrossRefGoogle Scholar
Sheppard, P. A. 1956 Airflow over mountains. Q. J. R. Met. Soc. 82, 528–529.CrossRefGoogle Scholar
Shishkina, O. D. 1996 Comparison of the drag coefficients of bodies moving in liquids with various stratification profiles. Fluid Dyn. 31, 484–489. [Transl. from Izv. Akad. Nauk Mekh. Zhidk. Gaza (4), 4–11.]CrossRefGoogle Scholar
Smith, R. B. 1980 Linear theory of stratified hydrostatic flow past an isolated mountain. Tellu. 32, 348–364.CrossRefGoogle Scholar
Smith, R. B. 1988 Linear theory of stratified flow past an isolated mountain in isosteric coordinates. J. Atmos. Sci. 45, 3889–3896.2.0.CO;2>CrossRefGoogle Scholar
Smith, R. B. 1989 a Mountain-induced stagnation points in hydrostatic flow. Tellus A 41, 270–274.CrossRefGoogle Scholar
Smith, R. B. 1989 b Hydrostatic airflow over mountains. Adv. Geophys. 31, 1–41.CrossRefGoogle Scholar
Smith, R. B. & Grønås, S. 1993 Stagnation points and bifurcation in 3-D mountain airflow. Tellus A 45, 28–43.CrossRefGoogle 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, 1154–1164.2.0.CO;2>CrossRefGoogle Scholar
Smolarkiewicz, P. K. & Rotunno, R. 1990 Low Froude number flow past three-dimensional obstacles. Part II: Upwind flow reversal zone. J. Atmos. Sci. 47, 1498–1511.2.0.CO;2>CrossRefGoogle Scholar
Snyder, W. H., Thompson, R. S., Eskridge, R. E., Lawson, R. E., Castro, I. P., Lee, J. T., Hunt, J. C. R. & Ogawa, Y. 1985 The structure of strongly stratified flow over hills: dividing-streamline concept. J. Fluid Mech. 152, 249–288.CrossRefGoogle Scholar
Spangler, T. C. 1987 Comparison of actual dividing-streamline heights to height predictions using the Froude number. J. Clim. Appl. Met. 26, 204–207.2.0.CO;2>CrossRefGoogle 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. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (6), 80–91.]CrossRefGoogle 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. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (3), 61–69.]CrossRefGoogle Scholar
Suzuki, M. & Kuwahara, K. 1992 Stratified flow past a bell-shaped hill. Fluid Dyn. Res. 9, 1–18.CrossRefGoogle 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. 1986 Vortex structure of a wake behind a sphere in a stratified fluid. J. Appl. Mech. Tech. Phys. 27, 190–196. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (2), 40–46.]CrossRefGoogle Scholar
Sysoeva, E. Ya. & Chashechkin, Yu. D. 1988 Spatial structure of a wake behind a sphere in a stratified liquid. J. Appl. Mech. Tech. Phys. 29, 655–660. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (5), 59–65.]CrossRefGoogle Scholar
Sysoeva, E. Ya. & Chashechkin, Yu. D. 1991 Vortex systems in the stratified wake of a sphere. Fluid Dyn. 26, 544–551. [Transl. from Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (4), 82–90.]CrossRefGoogle Scholar
Umeki, M. & Kambe, T. 1989 Stream patterns of an isothermal atmosphere over an isolated mountain. Fluid Dyn. Res. 5, 91–109.CrossRefGoogle 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. [Transl. from Zh. Prikl. Mekh. Tekh. Fiz. (2), 55–60.]CrossRefGoogle Scholar
Voisin, B. 1991 a Rayonnement des ondes internes de gravité. Application aux corps en mouvement. Ph.D. thesis, Université Pierre et Marie Curie, Paris.Google Scholar
Voisin, B. 1991 b Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources. J. Fluid Mech. 231, 439–480.CrossRefGoogle Scholar
Voisin, B. 1994 Internal wave generation in uniformly stratified fluids. Part 2. Moving point sources. J. Fluid Mech. 261, 333–374.CrossRefGoogle Scholar
Voisin, B. 1995 Internal wave generation by turbulent wakes. In Mixing in Geophysical Flows (ed. Redondo, J. M. & Métais, O.), pp. 291–301. CIMNE.Google Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243–293.CrossRefGoogle Scholar
Vosper, S. B. 2000 Three-dimensional numerical simulations of strongly stratified flow past conical orography. J. Atmos. Sci. 57, 3716–3739.2.0.CO;2>CrossRefGoogle Scholar
Vosper, S. B., Castro, I. P., Snyder, W. H. & Mobbs, S. D. 1999 Experimental studies of strongly stratified flow past three-dimensional orography. J. Fluid Mech. 390, 223–249.CrossRefGoogle Scholar
Warren, F. W. G. 1960 Wave resistance to vertical motion in a stratified fluid. J. Fluid Mech. 7, 209–229.CrossRefGoogle Scholar
Watson, G. N. 1944 A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Wu, T. Y.-T. 1965 Three-dimensional internal gravity waves in a stratified free-surface flow. Z. Angew. Math. Mech. Sonderh. 45, T194–T195.Google Scholar
Wurtele, M. G. 1957 The three-dimensional lee wave. Beitr. Phys. Atmos. 29, 242–252.Google Scholar
Wurtele, M. G., Sharman, R. D. & Datta, A. 1996 Atmospheric lee waves. Annu. Rev. Fluid Mech. 28, 429–476.CrossRefGoogle Scholar
Yih, C.-S. 1967 Equations governing steady three-dimensional large-amplitude motion of a stratified fluid. J. Fluid Mech. 29, 539–544.CrossRefGoogle Scholar
Zeytounian, R. Kh. 1969 Phénomènes d'ondes dans les écoulements stationnaires d'un fluide stratifié non visqueux. Première partie. Modèles théoriques. J. Méc. 8, 239–263.Google Scholar