### Abstract

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.

Original language | English (US) |
---|---|

Pages (from-to) | 565-679 |

Number of pages | 115 |

Journal | Journal of Statistical Physics |

Volume | 111 |

Issue number | 3-4 |

DOIs | |

State | Published - May 2003 |

### Fingerprint

### Keywords

- Closure approximations
- Direct interaction approximation
- Intermittency
- Passive scalar turbulence

### ASJC Scopus subject areas

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

### Cite this

**Closure Approximations for Passive Scalar Turbulence : A Comparative Study on an Exactly Solvable Model with Complex Features.** / Kramer, Peter R.; Majda, Andrew J.; Vanden-Eijnden, Eric.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 111, no. 3-4, pp. 565-679. https://doi.org/10.1023/A:1022837913026

}

TY - JOUR

T1 - Closure Approximations for Passive Scalar Turbulence

T2 - A Comparative Study on an Exactly Solvable Model with Complex Features

AU - Kramer, Peter R.

AU - Majda, Andrew J.

AU - Vanden-Eijnden, Eric

PY - 2003/5

Y1 - 2003/5

N2 - Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.

AB - Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.

KW - Closure approximations

KW - Direct interaction approximation

KW - Intermittency

KW - Passive scalar turbulence

UR - http://www.scopus.com/inward/record.url?scp=0037234415&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037234415&partnerID=8YFLogxK

U2 - 10.1023/A:1022837913026

DO - 10.1023/A:1022837913026

M3 - Article

AN - SCOPUS:0037234415

VL - 111

SP - 565

EP - 679

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -