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Particle dispersion by random waves in the rotating Boussinesq system

Published online by Cambridge University Press:  12 January 2011

MIRANDA HOLMES-CERFON ,
OLIVER BÜHLER  and
RAFFAELE FERRARI
MIRANDA HOLMES-CERFON*
Affiliation:
Center for Atmosphere Ocean Science at the Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
OLIVER BÜHLER
Affiliation:
Center for Atmosphere Ocean Science at the Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
RAFFAELE FERRARI
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present a theoretical and numerical study of horizontal particle dispersion due to random waves in the three-dimensional rotating and stratified Boussinesq system, which serves as a simple model to study the dispersion of tracers in the ocean by the internal wave field. Specifically, the effective one-particle diffusivity in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) is computed for a small-amplitude internal gravity wave field modelled as a stationary homogeneous and horizontally isotropic Gaussian random field whose frequency spectrum is bounded away from zero. Dispersion in this system does not arise simply because of a Stokes drift effect, as in the case of surface gravity waves, but in addition it is driven by the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A formula for the one-particle diffusivity as a function of the spectrum of the random wave field is presented. It is shown that this diffusivity is much smaller than might be expected from heuristic arguments based on the magnitude of the Stokes drift or the pseudomomentum. This appears to stem from certain incompressibility constraints for the Stokes drift and the second-order velocity field. Finally, the theory is applied to oceanic conditions described by a typical model wave spectrum, the Garrett–Munk spectrum, and also by detailed field observations from the North Atlantic tracer release experiment.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 150 - 175
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Balk, A. M. 2006 Wave turbulent diffusion due to the Doppler shift. J. Stat. Mech. P08018.Google Scholar
Balk, A. M., Falkovich, G. & Stepanov, M. G. 2004 Growth of density inhomogeneities in a flow of wave turbulence. Phys. Rev. Lett. 92, 244504.CrossRefGoogle Scholar
Balk, A. M. & McLaughlin, R. M. 1999 Passive scalar in a random wave field: the weak turbulence approach. Phys. Lett. A 256, 299–306.Google Scholar
Batchelor, G. 1952 Diffusion in a field of homogeneous turbulence. ii. The relative motion of particles. Proc. Camb. Phil. Soc. 48, 345–362.Google Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.Google Scholar
Bühler, O. & Holmes-Cerfon, M. 2009 Particle dispersion by random waves in rotating shallow water. J. Fluid Mech. 638, 5–26.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.CrossRefGoogle Scholar
Chertkov, M., Falkovich, G., Kolokolov, I. & Lebedev, V. 1995 Statistics of a passive scalar advected by a large-scale two-dimensional velocity field: analytic solution. Phys. Rev. E 51 (6), 5609–5627.Google Scholar
Davis, R. E. 1991 Observing the general circulation with floats. Deep-Sea Res. 28 (1), S531–S571.CrossRefGoogle Scholar
Herterich, K. & Hasselmann, K. 1982 The horizontal diffusion of tracers by surface waves. J. Phys. Oceanogr. 12, 704–712.Google Scholar
Ledwell, J., Watson, A. & Law, C. 1993 Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment. Nature 364, 701–703.Google Scholar
Ledwell, J., Watson, A. & Law, C. 1998 Mixing of a tracer in the pycnocline. J. Geophys. Res. 103 (C10), 21499–21529.Google Scholar
Joyce, T. M., Luyten, J. R., Kubrayakov, A., Bahr, F. B. & Pallant, J. 1998 Meso- to large-scale structure of subducting water in the subtropical gyre of the eastern North Atlantic Ocean. J. Phys. Oceanogr. 28, 40–61.2.0.CO;2>CrossRefGoogle Scholar
Kraichnan, R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 22–32.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Levine, M. D., Padman, L., Muench, R. D. & Morison, J. H. 1997 Internal waves and tides in the western Weddell Sea: observations from Ice Station Weddell. J. Geophys. Res. 102, 1073–1090.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Majda, A. J. & Kramer, P. R. 1999 Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314, 237–574.Google Scholar
Munk, W. 1981 Internal waves and small-scale processes. In Evolution of Physical Oceanography (ed. Warren, B. & Wunsch, C.), pp. 264–291. MIT Press.Google Scholar
Polzin, K. & Ferrari, R. 2004 Isopycnal dispersion in NATRE. J. Phys. Oceanogr. 34, 247–257.2.0.CO;2>CrossRefGoogle Scholar
Polzin, K. L., Toole, J. M. & Schmitt, R. W. 1995 Finescale parameterizations of turbulent dissipation. J. Phys. Oceanogr. 25, 306–328.2.0.CO;2>CrossRefGoogle Scholar
Sanderson, B. G. & Okubo, A. 1988 Diffusion by internal waves. J. Geophys. Res. 93, 3570–3582.CrossRefGoogle Scholar
Sawford, B. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33, 289–317.Google Scholar
Smith, K. S. & Ferrari, R. 2009 The production and dissipation of compensated thermohaline variance by mesoscale stirring. J. Phys. Oceanogr. 39, 2477–2501.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. 20, 196–212.Google Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid. Mech. 41, 375–404.CrossRefGoogle Scholar
Vucelja, M., Falkovich, G. & Fouxon, I. 2007 Clustering of matter in waves and currents. Phys. Rev. E 75, 065301.Google Scholar
Weichman, P. & Glazman, R. 2000 Passive scalar transport by travelling wave fields. J. Fluid Mech. 420, 147–200.Google Scholar
Weller, R. A., Furey, P. W., Spall, M. A. & Davis, R. E. 2004 The large-scale context for oceanic subduction in the Northeast Atlantic. Deep-Sea Res. I 51, 665–699.CrossRefGoogle Scholar
Young, W. R., Rhines, P. B. & Garrett, C. J. 1982 Shear-flow dispersion, internal waves and horizontal mixing in the ocean. J. Phys. Oceanogr. 12, 515–527.Google Scholar