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Weak compressibility of surface wave turbulence

  • MARIJA VUCELJA (a1) and ITZHAK FOUXON (a1) (a2)

We study the growth of small-scale inhomogeneities in the density of particles floating in weakly nonlinear small-amplitude surface waves. Despite the small amplitude, the accumulated effect of the long-time evolution may produce a strongly inhomogeneous distribution of the floaters: density fluctuations grow exponentially with a small but finite exponent. We show that the exponent is of sixth or higher order in wave amplitude. As a result, the inhomogeneities do not form within typical time scales of the natural environment. We conclude that the turbulence of surface waves is weakly compressible and alone it cannot be a realistic mechanism of the clustering of matter on liquid surfaces.

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Balk A. M. 2001 Anomalous diffusion of a tracer advected by wave turbulence. Phys. Lett. A 279, 370378.
Balk A. M., Falkovich G. & Stepanov M. G. 2004 Growth of density inhomogeneities in a flow of wave turbulence. Phys. Rev. Lett. 92, 244504.
Balkovsky E., Falkovich G. & Fouxon A. 2001 Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86, 27902793.
Bandi M. M., Goldburg W. I. & Cressman G. R. jr 2006 Measurement of entropy production rate in compressible turbulence. Europhys. Lett. 76, 595601.
Batchelor G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bec J., GawĘdzki K. & Horvai P. 2004 Multifractal clustering in compressible flows. Phys. Rev. Lett. 92, 224501.
Boffetta G., Davoudi J., Eckhardt B. & Schumacher J. 2006 a Lagrangian tracers on a surface flow: The role of time correlations. Phus. Rev. Lett. 93, 134501.
Boffetta G., Davoudi J. & de Lillo F. 2006 b Multifractal clustering of passive tracers on a surface flow. Europhys. Lett. 74, 6268.
Bohr T. & Hansen J. L. 1996 Chaotic particle motion under linear surface waves. Chaos 6, 554563.
Chelton D. B. & Eddy W. F. 1993 Statistics and Physical Oceanography. National Academy Press.
Crawford D. R., Saffman P. G. & Yuen H. C. 1980 Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion 2, 116.
Cressman J. R., Davoudi J., Goldburg W. & Schumacher J. 2004 Eulerian and lagrangian studies in surface flow turbulence. New J. Phys. 6, 53.
Cressman J. R. & Goldburg W. 2003 Compressible flow: Turbulence at the surface. J. Statist. Phys. 113, 875883.
Denissenko P., Falkovich G. & Lukaschuk S. 2006 How waves affect the distribution of particles that float on a liquid surface. Phys. Rev. Lett. 97, 244501.
Dorfman J. 1999 Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge University Press.
Eckhardt B. & Schumacher J. 2001 Turbulence and passive scalar transport in a free-slip surface. Phys. Rev. E 64, 016314.
Falkovich G. & Fouxon A. 2003 Entropy production away from the equilibrium.
Falkovich G. & Fouxon A. 2004 Entropy production and extraction in dynamical systems and turbulence. New J. Phys. 6, 50.
Falkovich G., GawĘdzki K. & Vergassola M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913975.
Herterich K. & Hasselmann K. 1982 The horizontal diffusion of tracers by surface waves. J. Phys. Oceanogr. 12, 704711.
Landau L. D. & Lifshitz E. M. 2000 Fluid Mechanics. Butterworth-Heinemann.
Longuet-Higgins M. S. 1963 The effect of non-linearities on statistical distributions in the theory of sea waves. J. Fluid. Mech. 17, 459480.
Nameson A., Antonsen T. & Ott E. 1996 Power law wave number spectra of fractal particle distributions advected by flowing fluids. Phys. Fluids 8, 24262434.
Ott E. 2002 Chaos in Dynamical Systems. Cambridge University Press.
Ramshankar R., Berlin D. & Gollub J. 1990 Transport by capillary waves. part i. particle trajectories. Phys. Fluids A 2, 19551965.
Reichl L. E. 1988 A Modern Course in Statistical Physics. University of Texas Press.
Ruelle D. 1996 Positivity of entropy production in nonequilibrium statistical mechanics. J. Statist Phys. 85, 123.
Ruelle D. 1997 Positivity of entropy production in the presence of a random thermostat. J. Statist Phys. 86, 935990.
Ruelle D. 1999 Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Statist Phys. 95, 393468.
Schroder E., Anderson J. S., Levinsen M. T. et al. 1996 Relative particle motion in capillary waves. Phys. Rev. Lett. 76, 47174720.
Sommerer J. C. 1996 Experimental evidence for power-law wave number spectra of fractal tracer distributions in a complicated surface flow. Phys. Fluids 8, 24412446.
Sommerer J. C. & Ott E. 1993 Particles floating on a moving fluid: A dynamically comprehensible physical fracta. Science 259, 335339.
Stepanov M. G. 2006 private communication.
Thorpe S. A. 2005 The Turbulent Ocean. Cambridge University Press.
Umeki M. 1992 Lagrangian motion of fluid particles induced by three-dimensional standing surface waves. Phys. Fluids A 4, 19681978.
Vucelja M., Falkovich G. & Fouxon I. 2007 Clustering of matter in waves and currents. Phys. Rev. E 75, 065301.
Weichman P. B. & Glazman R. E. 2000 Passive scalar transport by traveling wave fields. J. Fluid Mech. 420, 147200.
Yu L., Ott E. & Chen Q. 1991 Fractal distribution of floaters on a fluid surface and the transition to chaos for random maps. Physica D 53, 102124.
Zakharov V. E. 1966 Waves in nonlinear dispersive mediums. PhD thesis (in Russian). Institute for Nuclear Physics (Physics and Mathematics), Novosibirsk.
Zakharov V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid (note a sign misprint in (1.8)). J. Appl. Mech. Tech. Phys. 9, 190194.
Zakharov V. E. 1985 Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Rep. 129, 285366.
Zakharov V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth field. Eur. J. Mech. B/Fluids 18, 327344.
Zakharov V. E., L'vov V. & Falkovich G. 1992 Kolmogorov Spectra of Turbulence. Springer.
Zinn-Justin J. 2002 Quantum Field Theory and Critical Phenomena. Oxford University Press.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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