Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-28T03:38:11.443Z Has data issue: false hasContentIssue false

Particle dispersion by random waves in rotating shallow water

Published online by Cambridge University Press:  14 October 2009

OLIVER BÜHLER
Affiliation:
Center for Atmosphere Ocean Science at the Courant Institute of Mathematical Sciences New York University, New York, NY 10012, USA
MIRANDA HOLMES-CERFON*
Affiliation:
Center for Atmosphere Ocean Science at the Courant Institute of Mathematical Sciences New York University, New York, NY 10012, USA
*
Email address for correspondence: holmes@cims.nyu.edu

Abstract

We present a theoretical and numerical study of wave-induced particle dispersion due to random waves in the rotating shallow-water system, as part of an ongoing study of particle dispersion in the ocean. Specifically, the effective particle diffusivities in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) are computed for a small-amplitude wave field modelled as a stationary homogeneous isotropic Gaussian random field whose frequency spectrum is bounded away from zero. In this case, the leading-order diffusivity depends crucially on the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A closed-form analytic expression for the effective diffusivity is derived and carefully tested against numerical Monte Carlo simulations. The main conclusions are that Coriolis forces in shallow water invariably decrease the effective particle diffusivity and that there is a peculiar choking effect for the second-order particle flow in the limit of strong rotation.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Balk, A. M. 2006 Wave turbulent diffusion due to the doppler shift. J. Stat. Mech. P08018.CrossRefGoogle 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
Batchelor, G. 1952 Diffusion in a field of homogeneous turbulence. II. The relative motion of particles. Proc. Cambridge Philos. Soc. 48, 345362.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301343.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), 56095627.CrossRefGoogle ScholarPubMed
Herterich, K. & Hasselmann, K. 1982 The horizontal diffusion of tracers by surface waves. J. Phys. Oceanography 12, 704712.2.0.CO;2>CrossRefGoogle Scholar
Kraichnan, R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 2232.CrossRefGoogle Scholar
Ledwell, J., Watson, A. & Law, C. 1993 Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment. Nature 364, 701703.CrossRefGoogle Scholar
Ledwell, J., Watson, A. & Law, C. 1998 Mixing of a tracer in the pycnocline. J. Geophys. Res. 103 (C10), 2149921529.CrossRefGoogle Scholar
Majda, A. J. & Kramer, P. R. 1999 Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314, 237574.CrossRefGoogle Scholar
Polzin, K. & Ferrari, R. 2004 Isopycnal dispersion in NATRE. J. Phys. Oceanography 34, 247257.2.0.CO;2>CrossRefGoogle Scholar
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. London Ser. A 110, 709737.Google Scholar
Sanderson, B. G. & Okubo, A. 1988 Diffusion by internal waves. J. Geophys. Res. 93, 35703582.CrossRefGoogle Scholar
Sawford, B. 2001 Turbulent relative dispersion. Ann. Rev. Fluid Mech. 33, 289317.CrossRefGoogle Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. London Math. Soc. 20, 196212.Google Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Ann. Rev. Fluid. Mech. 41, 375404.CrossRefGoogle Scholar
Vucelja, M., Falkovich, G. & Fouxon, I. 2007 Clustering of matter in waves and currents. Phys. Rev. E 75 (065301).CrossRefGoogle ScholarPubMed
Weichman, P. & Glazman, R. 2000 Passive scalar transport by travelling wave fields. J. Fluid Mech. 420, 147200.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar
Yaglom, A. M. 1962 An Introduction to the Theory of Stationary Random Functions. Dover.Google Scholar
Yaglom, A. M. 1987 Correlation Theory of Stationary and Related Random Functions. Vol 1: Basic Results. Springer.Google Scholar