### Abstract

Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves like k_{0}(Pe)^{1/2} in the limit of large Péclet numbers where k_{0} is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

Original language | English (US) |
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Pages (from-to) | 339-391 |

Number of pages | 53 |

Journal | Communications in Mathematical Physics |

Volume | 138 |

Issue number | 2 |

DOIs | |

State | Published - May 1991 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows.** / Avellaneda, Marco; Majda, Andrew J.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 138, no. 2, pp. 339-391. https://doi.org/10.1007/BF02099496

}

TY - JOUR

T1 - An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

AU - Avellaneda, Marco

AU - Majda, Andrew J.

PY - 1991/5

Y1 - 1991/5

N2 - Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves like k0(Pe)1/2 in the limit of large Péclet numbers where k0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

AB - Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves like k0(Pe)1/2 in the limit of large Péclet numbers where k0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

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UR - http://www.scopus.com/inward/citedby.url?scp=0000705763&partnerID=8YFLogxK

U2 - 10.1007/BF02099496

DO - 10.1007/BF02099496

M3 - Article

VL - 138

SP - 339

EP - 391

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -