Chinese Annals of Mathematics, Series B

, Volume 30, Issue 5, pp 631–644 | Cite as

Homogenization theory for a replenishing passive scalar field

  • Peter R. KramerEmail author
  • Shane R. Keating


Homogenization theory provides a rigorous framework for calculating the effective diffusivity of a decaying passive scalar field in a turbulent or complex flow. The authors extend this framework to the case where the passive scalar fluctuations are continuously replenished by a source (and/or sink). The basic structure of the homogenized equations carries over, but in some cases the homogenized source can involve a non-trivial coupling of the velocity field and the source. The authors derive expressions for the homogenized source term for various multiscale source structures and interpret them physically.


Homogenization Turbulent transport Source Pumping 

2000 MR Subject Classification



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  1. [1]
    Ottino, J. M., Mixing, chaotic advection, and turbulence, Ann. Rev. Fluid Mech., 22, 1990, 207–254.CrossRefMathSciNetGoogle Scholar
  2. [2]
    Majda, A. J. and Kramer, P. R., Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Rep., 314(4–5), 1999, 237–574.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Papanicolaou, G. C. and Varadhan, S. R. S., Boundary value problems with rapidly oscillating random coefficients, Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory, Colloquia Mathematica Societatis János Bolyai, 2, J. Fritz, J. L. Lebowitz and D. Szasz (eds.), North-Holland, Amsterdam, 1979, 835–873.Google Scholar
  4. [4]
    Cioranescu, D. and Donato, P., An Introduction to Homogenization, Oxford University Press, New York, 1999.zbMATHGoogle Scholar
  5. [5]
    Bensoussan, A., Lions, J. L. and Papanicolaou, G., Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, 5, North-Holland, Amsterdam, 1978.Google Scholar
  6. [6]
    Jikov, V. V., Kozlov, S. M. and Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, 55–85.Google Scholar
  7. [7]
    Avellaneda, M. and Vergassola, M., Stieltjes integral representation of effective diffusivities in timedependent flows, Phys. Rev. E, 52(3), 1995, 3249–3251.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Avellaneda, M. and Majda, A. J., Stieltjes integral representation and effective diffusivity bounds for turbulent transport, Phys. Rev. Lett., 62(7), 1989, 753–755.CrossRefGoogle Scholar
  9. [9]
    Fannjiang, A. and Papanicolaou, G., Diffusion in turbulence, Prob. Th. Rel. Fields, 105(3), 1996, 279–334.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Oelschläger, K., Homogenization of a diffusion process in a divergence-free random field, Ann. Probab., 16(3), 1988, 1084–1126.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Olla, S., Homogenization of Diffusion Processes in Random Fields, Lecture Notes at Ecole Polytechnique, Ecole Polytechnique, Paris, 1994.Google Scholar
  12. [12]
    Plasting, S. C. and Young, W. R., A bound on scalar variance for the advection-diffusion equation, J. Fluid Mech., 552, 2006, 289–298.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Abraham, E. R. and Bowen, M. M., Chaotic stirring by a mesoscale surfaceocean flow, Chaos, 12(2), 2002, 373–381.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Martin, A. P., Phytoplankton patchiness: the role of lateral stirring and mixing, Prog. Oceanogr., 57, 2003, 125–174.CrossRefGoogle Scholar
  15. [15]
    Shaw, T. A., Thiffeault, J.-L. and Doering, C. R., Stirring up trouble: multi-scale mixing measures for steady scalar sources, Phys. D, 231(2), 2007, 143–164.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Avellaneda, M. and Majda, A. J., An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows, Comm. Math. Phys., 138, 1991, 339–391.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Lin, Z., Bod’ová, K. and Doering, C. R., Measures of mixing and effective diffusion scalings, 2009, preprint.Google Scholar
  18. [18]
    Koch, D. L. and Brady, J. F., The symmetry properties of the effective diffusivity tensor in anisotropic porous media, Phys. Fluids, 30(3), 1987, 642–650.zbMATHCrossRefGoogle Scholar
  19. [19]
    Middleton, J. F. and Loder, J. W., Skew fluxes in polarized wave fields, J. Phys. Oceanogr., 19, 1989, 68–76.CrossRefGoogle Scholar
  20. [20]
    Moffatt, H. K., Transport effects associated with turbulence with particular attention to the influence of helicity, Rep. Prog. Phys., 46, 1983, 621–664.CrossRefGoogle Scholar
  21. [21]
    Griffies, S. M., The Gent-McWilliams skew flux, J. Phys. Oceanogr., 28, 1998, 831–841.CrossRefGoogle Scholar
  22. [22]
    Gent, P. R., Willebrand, J., McDougall, T. J., et al, Parameterizing eddy-induced tracer transports in ocean circulation models, J. Phys. Oceanogr., 25, 1995, 463–474.CrossRefGoogle Scholar
  23. [23]
    Canuto, V. M., The physics of subgrid scales in numerical simulations of stellar convection: are they dissipative, advective, or diffusive? Astrophys. J. Lett., 541, 2000, L79–L82.CrossRefGoogle Scholar
  24. [24]
    Keating, S. R. and Kramer, P. R., A homogenization perspective on mixing efficiency measures, 2009, preprint.Google Scholar
  25. [25]
    Lin, C. C. and Segel, L. A., Mathematics Applied to Deterministic Problems in the Natural Sciences, With material on elasticity by G. H. Handelman, With a foreword by Robert E. O’Malley, Jr., SIAM, Philadelphia, 1988.Google Scholar
  26. [26]
    Folland, G. B., Introduction to Partial Differential Equations, Second Edition, Princeton University Press, Princeton, 1995.zbMATHGoogle Scholar
  27. [27]
    Goudon, T. and Poupaud, F., Homogenization of transport equations: weak mean field approximation, SIAM J. Math. Anal., 36(3), 2005, 856–881.CrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA
  2. 2.Center for Atmosphere-Ocean ScienceCourant Institute of Mathematical Sciences, New York UniversityNew YorkUSA

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