Editorial Type:
Article
Article Type:
Research Article

The Effect of Bubbles on Internal Waves

R. H. J. Grimshaw Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom

Search for other papers by R. H. J. Grimshaw in
Current site
Google Scholar
PubMed Close
and
K. R. Khusnutdinova Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom, and Institute of Mechanics, Ufa Branch, Russian Academy of Sciences, Ufa, Russia

Search for other papers by K. R. Khusnutdinova in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Feb 2004
DOI:
https://doi.org/10.1175/1520-0485(2004)034<0477:TEOBOI>2.0.CO;2
Page(s):
477–489
Restricted access

Abstract

The influence of gas bubbles on the properties of internal waves in a continuously stratified fluid is studied in the framework of a two-dimensional model of a diluted locally monodisperse mixture of an incompressible fluid with gas bubbles. The model takes into account the depth dependence of the void fraction of the bubbles, surface tension on the walls of the bubbles, and an effective viscosity, which accounts for the fluid viscosity, thermal damping, and other dissipative mechanisms. It is shown that bubbles, when present in the upper part of the ocean, change the value of the buoyancy frequency Nl in the absence of bubbles, replacing it with an effective value N, where N2 ≈ N2l + gαg0(lnn0)z (αg0 is the void fraction and n0 is the number density of the bubbles). First, plane linear waves in a uniformly stratified Boussinesq fluid are considered, and it is shown that there are two classes of plane waves. One class, the “bubble” wave, may propagate with frequencies higher than the effective buoyancy frequency N, and the other class is a modified internal wave, whose frequency is less than the effective buoyancy frequency, with a finite gap in the spectrum existing for all wavenumbers. The effective viscosity introduces a damping of both modes and has a greater effect on the bubble mode. Then the dispersion relation for waves propagating horizontally in the oceanic waveguide is obtained, for both the case in which the fluid is uniformly stratified and contains bubbles and the case in which the bubbles are confined to a thin nearly homogeneous upper layer.

Corresponding author address: Prof. Roger H. J. Grimshaw, Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom. Email: r.h.j.grimshaw@lboro.ac.uk

Abstract

The influence of gas bubbles on the properties of internal waves in a continuously stratified fluid is studied in the framework of a two-dimensional model of a diluted locally monodisperse mixture of an incompressible fluid with gas bubbles. The model takes into account the depth dependence of the void fraction of the bubbles, surface tension on the walls of the bubbles, and an effective viscosity, which accounts for the fluid viscosity, thermal damping, and other dissipative mechanisms. It is shown that bubbles, when present in the upper part of the ocean, change the value of the buoyancy frequency Nl in the absence of bubbles, replacing it with an effective value N, where N2 ≈ N2l + gαg0(lnn0)z (αg0 is the void fraction and n0 is the number density of the bubbles). First, plane linear waves in a uniformly stratified Boussinesq fluid are considered, and it is shown that there are two classes of plane waves. One class, the “bubble” wave, may propagate with frequencies higher than the effective buoyancy frequency N, and the other class is a modified internal wave, whose frequency is less than the effective buoyancy frequency, with a finite gap in the spectrum existing for all wavenumbers. The effective viscosity introduces a damping of both modes and has a greater effect on the bubble mode. Then the dispersion relation for waves propagating horizontally in the oceanic waveguide is obtained, for both the case in which the fluid is uniformly stratified and contains bubbles and the case in which the bubbles are confined to a thin nearly homogeneous upper layer.

Corresponding author address: Prof. Roger H. J. Grimshaw, Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom. Email: r.h.j.grimshaw@lboro.ac.uk

Save
Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Akhatov, I. Sh, 1989: The shear instability in stratified viscoelastic and liquid-gas media. J. Appl. Math. Mech., 53 , 630–635.

  • Akhatov, I. Sh, R. I. Nigmatulin, D. B. Khismatullin, and K. R. Khusnutdinova, 1998: Resonant interaction of long and short pressure waves in bubbly liquids. Proc. Third Int. Conf. on Multiphase Flow, Lyon, France, CD-ROM, Paper 361.

    • Search Google Scholar
    • Export Citation

  • Arnold, V. I., 1989: Mathematical Methods of Classical Mechanics. Springer, 508 pp.

  • Benney, D. J., 1976: Significant interactions between small and large scale surface waves. Stud. Appl. Math., 55 , 93–106.

  • Benney, D. J., 1977: A general theory for interactions between short and long waves. Stud. Appl. Math., 56 , 81–94.

  • Brennen, C. E., 1995: Cavitation and Bubble Dynamics. Oxford University Press, 282 pp.

  • Brennen, C. E., 1998: Cloud cavitation: Observations, calculations and shock waves. Multiphase Sci. Technol., 10 , 303–321.

  • Buckingham, M. J., 1997: Sound speed and void fraction profiles in the sea surface bubble layer. Appl. Acoust., 51 , 225–250.

  • Bullock, G. N., A. R. Crawford, P. J. Hewson, M. J. A. Walkden, and P. A. D. Bird, 2001: The influence pf air and scale on wave impact pressures. Coastal Eng., 42 , 291–312.

    • Search Google Scholar
    • Export Citation

  • Crawford, G. B., and D. M. Farmer, 1987: On the spatial distribution of ocean bubbles. J. Geophys. Res., 92 , 8231–8243.

  • d'Agostino, L., and C. E. Brennen, 1989: Linearized dynamics of spherical bubble clouds. J. Fluid Mech., 199 , 155–176.

  • Dean, G. B., and M. D. Stokes, 1999: Air entraiment processes and bubble size distributions in the surf zone. J. Phys. Oceanogr., 29 , 1393–1403.

    • Search Google Scholar
    • Export Citation

  • Druzhinin, O. A., L. A. Ostrovsky, and A. Prosperetti, 1996: Low-frequency acoustic wave generation in a resonant bubble-layer. J. Acoust. Soc. Amer., 100 , 3570.

    • Search Google Scholar
    • Export Citation

  • Farmer, D. M., and D. D. Lemon, 1984: The influence of bubbles on ambient noise in the ocean at high wind speeds. J. Phys. Oceanogr., 14 , 1762–1778.

    • Search Google Scholar
    • Export Citation

  • Farmer, D. M., and D. D. Lemon, 1989: Waveguide propagation of ambient sound in the ocean-surface bubble layer. J. Acoust. Soc. Amer., 86 , 1897–1908.

    • Search Google Scholar
    • Export Citation

  • Farmer, D. M., D. D. Lemon, and A. D. Booth, 1998: A free-flooding acoustical resonator for measurement of bubble size distributions. J. Atmos. Oceanic Technol., 15 , 1132–1146.

    • Search Google Scholar
    • Export Citation

  • Farmer, D. M., D. D. Lemon, and M. Li, 1999: Wave breaking, turbulence and bubble distributions in the ocean surface layer. Proceedings of the Wind-Driven Air–Sea Interface, M. L. Banner, Ed., The University of New South Wales, 187–192.

    • Search Google Scholar
    • Export Citation

  • Garrett, C., M. Li, and D. M. Farmer, 2000: The connection between bubble size and energy dissipation rates in the upper ocean. J. Phys. Oceanogr., 30 , 2163–2171.

    • Search Google Scholar
    • Export Citation

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Grimshaw, R. H. J., 1977: The modulation of an internal gravity wave packet and the resonance with the mean motion. Stud. Appl. Math., 56 , 241–266.

    • Search Google Scholar
    • Export Citation

  • Gumerov, N. A., 1992: Quasi-monochromatic weakly non-linear waves is a low-dissipative bubbly medium. J. Appl. Math. Mech., 56 , 58–67.

    • Search Google Scholar
    • Export Citation

  • Herring, C., 1941: Theory of the pulsations of the gas bubble produced by an underwater explosion. OSRD Rep. 236.

  • Hickling, R., and M. S. Plesset, 1964: Collapse and rebound of a spherical bubble in water. Phys. Fluids, 7 , 7–14.

  • Iordansky, S. V., 1960: Equations of motion of liquid containing gas bubbles (in Russian). Zh. Prikl. Mekh. Tekh. Fiz., 3 , 102–110.

  • Johnson, B. D., and R. C. Cooke, 1979: Bubble populations and spectra in coastal waters: A photographic approach. J. Geophys. Res., 84 , 3761–3766.

    • Search Google Scholar
    • Export Citation

  • Karpov, S., A. Prosperetti, and L. Ostrovsky, 2003: Nonlinear wave interactions in bubble layers. J. Acoust. Soc. Amer., 113 , 1304–1316.

    • Search Google Scholar
    • Export Citation

  • Kedrinsky, V. K., 1968: Propagation of small disturbances in liquid containing gas bubbles (in Russian). Zh. Prikl. Mekh. Tekh. Fiz., 4 , 29–34.

    • Search Google Scholar
    • Export Citation

  • Khismatullin, D. B., and I. Sh Akhatov, 2001: Sound–ultrasound interaction in bubbly fluids: Theory and possible applications. Phys. Fluids, 13 , 3582–3598.

    • Search Google Scholar
    • Export Citation

  • Knapp, R. T., J. W. Dayly, and F. G. Hammit, 1970: Cavitation. McGraw-Hill, 578 pp.

  • Kogarko, B. S., 1961: On the model of cavitating liquid (in Russian). Dokl. AN SSSR, 137 , 1331–1333.

  • Kolovaev, P. A., 1976: Investigation of the concentration and statistical size distribution of wind-produced bubbles in the near surface ocean layer. Oceanology, 15 , 659–661.

    • Search Google Scholar
    • Export Citation

  • Kumar, S., and C. E. Brennen, 1993: Some nonlinear interactive effects in bubbly clouds. J. Fluid Mech., 253 , 565–591.

  • Kutateladze, S., and V. E. Nakoryakov, 1984: Heat and Mass-Transfer and Waves in Gas–Liquid Systems (in Russian). Nauka, 301 pp.

  • Mandelshtam, L. I., 1972: Lectures on the Theory of Oscillations (in Russian). Nauka, 470 pp.

  • Medwin, H., 1974: Acoustic fluctuations due to microbubbles in the near-surface ocean. J. Acoust. Soc. Amer., 56 , 1100–1104.

  • Medwin, H., 1977: In situ acoustic measurements of microbubbles at sea. J. Geophys. Res., 82 , 971–976.

  • Melville, W. K., 1996: The role of surface-wave breaking in air–sea interaction. Annu. Rev. Fluid Mech., 28 , 279–321.

  • Minnaert, M., 1933: On musical air bubbles and the sounds of running water. Philos. Mag., 16 , 235–248.

  • Nakoryakov, V. E., V. G. Pokusaev, and I. R. Shreiber, 1983: Propagation of Waves in Gas- And Vapour-Liquid Media (in Russian). Institute of Thermophysics, 280 pp.

    • Search Google Scholar
    • Export Citation

  • Nigmatulin, R. I., 1991a: Dynamics of Multiphase Media. Vol. 1. Hemisphere Publishers, 532 pp.

  • Nigmatulin, R. I., 1991b: Dynamics of Multiphase Media. Vol. 2. Hemisphere Publishers, 388 pp.

  • Ostrovsky, L. A., A. M. Sutin, I. A. Soustova, A. I. Matveyev, and A. I. Potapov, 1998: Nonlinear, low-frequency sound generation in a bubble layer: Theory and laboratory experiment. J. Acoust. Soc. Amer., 104 , 722–726.

    • Search Google Scholar
    • Export Citation

  • Plesset, M. S., and A. Prosperetti, 1977: Bubble dynamics and cavitation. Annu. Rev. Fluid Mech., 9 , 145–185.

  • Prosperetti, A., 1994: Bubble dynamics: Some things we did not know 10 years ago. Bubble Dynamics and Interface Phenomena, J. Blake et al., Eds., Kluwer, 3–16.

    • Search Google Scholar
    • Export Citation

  • Rayleigh, Lord, 1917: On the pressure developed in a liquid during the collapse of a spherical cavity. Philos. Mag., 34 , 94–98.

  • Reisman, G. E., Y. C. Wang, and C. E. Brennen, 1998: Observation of shock waves in cloud cavitation. J. Fluid Mech., 355 , 255–283.

  • Scott, J. C., 1975: The role of salt in whitecap persistence. Deep-Sea Res., 22 , 653–657.

  • Terrill, E. J., and W. K. Melville, 1999: Field measurements of bubble size distributions in the upper mixed layer. Proceedings of the Wind-Driven Air–Sea interface, M. L. Banner, Ed., The University of New South Wales, 247–255.

    • Search Google Scholar
    • Export Citation

  • Thorpe, S. A., 1982: On the clouds of bubbles formed by breaking wind-waves in deep water, and their role in air–sea gas transfer. Philos. Trans. Roy. Soc. London, 304A , 155–210.

    • Search Google Scholar
    • Export Citation

  • Thorpe, S. A., 1984a: On the determination of Kυ in the near-surface ocean from acoustic measurements of bubbles. J. Phys. Oceanogr., 14 , 855–863.

    • Search Google Scholar
    • Export Citation

  • Thorpe, S. A., 1984b: A model of the turbulent diffusion of bubbles below the sea surface. J. Phys. Oceanogr., 14 , 841–853.

  • Thorpe, S. A., 1995: Dynamical processes of transfer at the sea surface. Progress in Oceanography, Vol. 35, Pergamon, 315–352.

  • Thorpe, S. A., and A. J. Hall, 1983: The characteristics of breaking waves, bubble clouds, and near-surface currents observed using side-scan sonar. Cont. Shelf Res., 1 , 353–384.

    • Search Google Scholar
    • Export Citation

  • Thorpe, S. A., M. B. Belloul, and A. J. Hall, 1987: Internal waves and whitecaps. Nature, 330 , 740–742.

  • Trilling, L., 1952: The collapse and rebound of a gas bubble. J. Appl. Phys., 23 , 14–17.

  • Urick, R. J., 1983: Principles of Underwater Sound. McGraw-Hill, 384 pp.

  • van Wijngaarden, L., 1968: On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech., 33 , 465–474.

  • Woolf, D. K., and S. A. Thorpe, 1991: Bubbles and the air–sea exchange of gases in near saturation conditions. J. Mar. Res., 49 , 435–466.

    • Search Google Scholar
    • Export Citation

  • Zakharov, V. E., 1972: Collapse of Langmuir waves. Sov. Phys. JETP, 72 , 908.

  • Zedel, L., and D. Farmer, 1991: Organized structures in subsurface bubble clouds: Langmuir circulation in the open ocean. J. Geophys. Res., 96 , (C5),. 8889–8900.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 536 332 41
PDF Downloads 121 29 3