Skip to main content

Part of the book series: Lecture Notes in Physics ((LNP,volume 744))

Abstract

I review the local theory of mixing, which focuses on infinitesimal blobs of scalar being advected and stretched by a random velocity field. An advantage of this theory is that it provides elegant analytical results. A disadvantage is that it is highly idealised. Nevertheless, it provides insight into the mechanism of chaotic mixing and the effect of random fluctuations on the rate of decay of the concentration field of a passive scalar.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • G. K. Batchelor: Small-scale variation of convected quantities like temperature in turbulent fluid: part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 134 (1959)

    Article  Google Scholar 

  • R. H. Kraichnan: Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945 (1968)

    Article  Google Scholar 

  • R. H. Kraichnan: Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737 (1974)

    Article  Google Scholar 

  • Y. B. Zeldovich, A. A. Ruzmaikin, S. A. Molchanov and D. D. Sokoloff: Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 1 (1984)

    Article  Google Scholar 

  • E. Ott and T. M. Antonsen, Jr.: Fractal measures of passively convected vector fields and scalar gradients in chaotic fluid flows. Phys. Rev. A 39, 3660 (1989)

    Article  Google Scholar 

  • T. M. Antonsen, Jr and E. Ott: Multifractal power spectra of passive scalars convected by chaotic fluid flows. Phys. Rev. A 44, 851 (1991)

    Article  Google Scholar 

  • T. M. Antonsen, Jr, Z. Fan and E. Ott: k spectrum of passive scalars in Lagrangian chaotic fluid flows. Phys. Rev. Lett. 75, 1751 (1995)

    Article  Google Scholar 

  • T. M. Antonsen, Jr, Z. Fan, E. Ott and E. Garcia-Lopez: The role of chaotic orbits in the determination of power spectra. Phys. Fluids 8, 3094 (1996)

    Article  Google Scholar 

  • B. I. Shraiman and E. D. Siggia: Lagrangian path integrals and fluctuations in random flow. Phys. Rev. E 49, 2912 (1994)

    Article  Google Scholar 

  • M. Chertkov, G. Falkovich I. Kolokolov and V. Lebedev: Statistics of a passive scalar advected by a large-scale two-dimensional velocity field: analytic solution. Phys. Rev. E 51, 5609 (1995)

    Article  Google Scholar 

  • M. Chertkov, I. Kolokolov and M. Vergassola: Inverse cascade and intermittency of passive scalar in one-dimensional smooth flow. Phys. Rev. E 56, 5483 (1997)

    Article  Google Scholar 

  • B. I. Shraiman and E. D. Siggia: Scalar turbulence. Nature 405, 639 (2000)

    Article  Google Scholar 

  • G. Falkovich, K. GawÈ©dzki and M. Vergassola: Particles and fields in turbulence. Rev. Mod. Phys. 73, 913 (2001)

    Article  Google Scholar 

  • J.-L. Thiffeault: The strange eigenmode in Lagrangian coordinates. Chaos 14, 531 (2004)

    Article  Google Scholar 

  • E. Balkovsky and A. Fouxon: Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem. Phys. Rev. E 60, 4164 (1999)

    Article  Google Scholar 

  • R. S. Ellis: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)

    Google Scholar 

  • A. Schwartz and A. Weiss: Large Deviations for Performance Analysis. Chapman & Hall, London (1995)

    Google Scholar 

  • V. I. Oseledec: A multiplicative theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197 (1968)

    Google Scholar 

  • E. Ott: Chaos in Dynamical Systems. Cambridge University Press, Cambridge, UK (1994)

    Google Scholar 

  • A. Fouxon: Evolution of a scalar gradient’s probability density function in a random flow. Phys. Rev. E 58, 4019 (1998)

    Article  Google Scholar 

  • D. T. Son: Turbulent decay of a passive scalar in the Batchelor limit: exact results from a quantum-mechanical approach. Phys. Rev. E 59, R3811 (1999)

    Google Scholar 

  • A. D. Stroock, S. K. W. Dertinger, A. Ajdari, I. Mezić, H. A. Stone and G. M. Whitesides: Chaotic mixer for microchannels. Science 295, 647 (2002)

    Article  Google Scholar 

  • S. Hong, J.-L. Thiffeault, L. Fréchette and V. Modi: International Mechanical Engineering Congress & Exposition, Washington, DC. American Society of Mechanical Engineers, New York (2003)

    Google Scholar 

  • M. A. Ewart and J.-L. Thiffeault: A simple model for a microchannel mixer. in preparation (2005)

    Google Scholar 

  • J.-P. Eckmann and D. Ruelle: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985)

    Article  Google Scholar 

  • D. R. Fereday and P. H. Haynes: Scalar decay in two-dimensional chaotic advection and Batchelor-regime turbulence. Phys. Fluids 16, 4359 (2004)

    Article  Google Scholar 

  • R. T. Pierrehumbert: Tracer microstructure in the large-eddy dominated regime. Chaos Solitons Fractals 4, 1091 (1994)

    Article  Google Scholar 

  • D. Rothstein, E. Henry and J. P. Gollub: Persistent patterns in transient chaotic fluid mixing. Nature 401, 770 (1999)

    Article  Google Scholar 

  • D. R. Fereday, P. H. Haynes, A. Wonhas and J. C. Vassilicos: Scalar variance decay in chaotic advection and Batchelor-regime turbulence. Phys. Rev. E 65, 035301(R) (2002)

    Article  Google Scholar 

  • J. Sukhatme and R. T. Pierrehumbert: Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains: from non-self-similar probability distribution functions to self-similar eigenmodes. Phys. Rev. E 66, 056032 (2002)

    Article  Google Scholar 

  • A. Wonhas and J. C. Vassilicos: Mixing in fully chaotic flows. Phys. Rev. E 66, 051205 (2002)

    Article  Google Scholar 

  • A. Pikovsky and O. Popovych: Persistent patterns in deterministic mixing flows. Europhys. Lett. 61, 625 (2003)

    Article  Google Scholar 

  • J.-L. Thiffeault and S. Childress: Chaotic mixing in a torus map. Chaos 13, 502 (2003)

    Article  Google Scholar 

  • W. Liu and G. Haller: Strange eigenmodes and decay of variance in the mixing of diffusive tracers. Physica D 188, 1 (2004)

    Article  Google Scholar 

  • A. Schekochihin, P. H. Haynes and S. C. Cowley: Diffusion of passive scalar in a finite-scale random flow. Phys. Rev. E 70, 046304 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Thiffeault, JL. (2008). Scalar Decay in Chaotic Mixing. In: Weiss, J.B., Provenzale, A. (eds) Transport and Mixing in Geophysical Flows. Lecture Notes in Physics, vol 744. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75215-8_1

Download citation

Publish with us

Policies and ethics