Skip to main content

Particle dispersion by random waves in rotating shallow water


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.

Corresponding author
Email address for correspondence:
Hide All
Balk A. M. 2006 Wave turbulent diffusion due to the doppler shift. J. Stat. Mech. P08018.
Balk A. M., Falkovich G. & Stepanov M. G. 2004 Growth of density inhomogeneities in a flow of wave turbulence. Phys. Rev. Lett. 92 (244504).
Batchelor G. 1952 Diffusion in a field of homogeneous turbulence. II. The relative motion of particles. Proc. Cambridge Philos. Soc. 48, 345362.
Bühler O. & McIntyre M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301343.
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.
Herterich K. & Hasselmann K. 1982 The horizontal diffusion of tracers by surface waves. J. Phys. Oceanography 12, 704712.
Kraichnan R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 2232.
Ledwell J., Watson A. & Law C. 1993 Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment. Nature 364, 701703.
Ledwell J., Watson A. & Law C. 1998 Mixing of a tracer in the pycnocline. J. Geophys. Res. 103 (C10), 2149921529.
Majda A. J. & Kramer P. R. 1999 Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314, 237574.
Polzin K. & Ferrari R. 2004 Isopycnal dispersion in NATRE. J. Phys. Oceanography 34, 247257.
Richardson L. F. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. London Ser. A 110, 709737.
Sanderson B. G. & Okubo A. 1988 Diffusion by internal waves. J. Geophys. Res. 93, 35703582.
Sawford B. 2001 Turbulent relative dispersion. Ann. Rev. Fluid Mech. 33, 289317.
Taylor G. I. 1921 Diffusion by continuous movements. Proc. London Math. Soc. 20, 196212.
Toschi F. & Bodenschatz E. 2009 Lagrangian properties of particles in turbulence. Ann. Rev. Fluid. Mech. 41, 375404.
Vucelja M., Falkovich G. & Fouxon I. 2007 Clustering of matter in waves and currents. Phys. Rev. E 75 (065301).
Weichman P. & Glazman R. 2000 Passive scalar transport by travelling wave fields. J. Fluid Mech. 420, 147200.
Whitham G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.
Yaglom A. M. 1962 An Introduction to the Theory of Stationary Random Functions. Dover.
Yaglom A. M. 1987 Correlation Theory of Stationary and Related Random Functions. Vol 1: Basic Results. Springer.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 17 *
Loading metrics...

Abstract views

Total abstract views: 85 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th November 2017. This data will be updated every 24 hours.