### Abstract

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With ε the spectral parameter, for -∞<ε<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for ε<2 to anomalous behavior for ε with 2<ε<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for ε with ε<2 to an explicit family with a non-Gaussian covariance for ε with 2<ε<4. These non-Gaussian distributions have tails that are broader than Gaussian as ε varies with 2<ε<4 and behave for large values like exp(-C_{c}|x|^{4-ε}), with C_{c} an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.

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

Pages (from-to) | 1153-1165 |

Number of pages | 13 |

Journal | Journal of Statistical Physics |

Volume | 75 |

Issue number | 5-6 |

DOIs | |

State | Published - Jun 1994 |

### Fingerprint

### Keywords

- long-range correlations
- renormalization
- shear layers
- Turbulent diffusion

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*75*(5-6), 1153-1165. https://doi.org/10.1007/BF02186761

**Random shearing direction models for isotropic turbulent diffusion.** / Majda, Andrew J.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 75, no. 5-6, pp. 1153-1165. https://doi.org/10.1007/BF02186761

}

TY - JOUR

T1 - Random shearing direction models for isotropic turbulent diffusion

AU - Majda, Andrew J.

PY - 1994/6

Y1 - 1994/6

N2 - Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With ε the spectral parameter, for -∞<ε<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for ε<2 to anomalous behavior for ε with 2<ε<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for ε with ε<2 to an explicit family with a non-Gaussian covariance for ε with 2<ε<4. These non-Gaussian distributions have tails that are broader than Gaussian as ε varies with 2<ε<4 and behave for large values like exp(-Cc|x|4-ε), with Cc an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.

AB - Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With ε the spectral parameter, for -∞<ε<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for ε<2 to anomalous behavior for ε with 2<ε<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for ε with ε<2 to an explicit family with a non-Gaussian covariance for ε with 2<ε<4. These non-Gaussian distributions have tails that are broader than Gaussian as ε varies with 2<ε<4 and behave for large values like exp(-Cc|x|4-ε), with Cc an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.

KW - long-range correlations

KW - renormalization

KW - shear layers

KW - Turbulent diffusion

UR - http://www.scopus.com/inward/record.url?scp=21344477476&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21344477476&partnerID=8YFLogxK

U2 - 10.1007/BF02186761

DO - 10.1007/BF02186761

M3 - Article

VL - 75

SP - 1153

EP - 1165

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -