### Abstract

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family of N- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter e{open}, which measures the strength of the infrared divergence: for e{open}<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for e{open}>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.

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

Pages (from-to) | 515-542 |

Number of pages | 28 |

Journal | Journal of Statistical Physics |

Volume | 73 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 1993 |

### Fingerprint

### Keywords

- inertial range renormalization
- long-range correlations
- quantum mechanical analogy
- Turbulent diffusion

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*73*(3-4), 515-542. https://doi.org/10.1007/BF01054338

**Explicit inertial range renormalization theory in a model for turbulent diffusion.** / Majda, Andrew J.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 73, no. 3-4, pp. 515-542. https://doi.org/10.1007/BF01054338

}

TY - JOUR

T1 - Explicit inertial range renormalization theory in a model for turbulent diffusion

AU - Majda, Andrew J.

PY - 1993/11

Y1 - 1993/11

N2 - The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family of N- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter e{open}, which measures the strength of the infrared divergence: for e{open}<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for e{open}>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.

AB - The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family of N- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter e{open}, which measures the strength of the infrared divergence: for e{open}<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for e{open}>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.

KW - inertial range renormalization

KW - long-range correlations

KW - quantum mechanical analogy

KW - Turbulent diffusion

UR - http://www.scopus.com/inward/record.url?scp=21344490696&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21344490696&partnerID=8YFLogxK

U2 - 10.1007/BF01054338

DO - 10.1007/BF01054338

M3 - Article

VL - 73

SP - 515

EP - 542

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -