### Abstract

The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large-deviation problem for the scalar field into a small-deviation, or small-ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here such a methodology is applied to a model proposed by Majda consisting of a white-in-time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non-Gaussian invariant measure for the (reduced) scalar field is derived, and, in particular, it is shown that the one-point distribution of the scalar has stretched exponential tails, with a stretching exponent depending on the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one-to-one correspondence with an exponent measuring time persistence effects in the velocity field.

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

Pages (from-to) | 1146-1167 |

Number of pages | 22 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 54 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2001 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Non-Gaussian invariant measures for the majda model of decaying turbulent transport.** / Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Non-Gaussian invariant measures for the majda model of decaying turbulent transport

AU - Vanden Eijnden, Eric

PY - 2001/9

Y1 - 2001/9

N2 - The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large-deviation problem for the scalar field into a small-deviation, or small-ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here such a methodology is applied to a model proposed by Majda consisting of a white-in-time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non-Gaussian invariant measure for the (reduced) scalar field is derived, and, in particular, it is shown that the one-point distribution of the scalar has stretched exponential tails, with a stretching exponent depending on the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one-to-one correspondence with an exponent measuring time persistence effects in the velocity field.

AB - The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large-deviation problem for the scalar field into a small-deviation, or small-ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here such a methodology is applied to a model proposed by Majda consisting of a white-in-time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non-Gaussian invariant measure for the (reduced) scalar field is derived, and, in particular, it is shown that the one-point distribution of the scalar has stretched exponential tails, with a stretching exponent depending on the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one-to-one correspondence with an exponent measuring time persistence effects in the velocity field.

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U2 - 10.1002/cpa.3001

DO - 10.1002/cpa.3001

M3 - Article

VL - 54

SP - 1146

EP - 1167

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 9

ER -