Random shearing direction models for isotropic turbulent diffusion

Andrew J. Majda

Research output: Contribution to journalArticle

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(-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.

Original languageEnglish (US)
Pages (from-to)1153-1165
Number of pages13
JournalJournal of Statistical Physics
Volume75
Issue number5-6
DOIs
StatePublished - Jun 1994

Fingerprint

Turbulent Diffusion
turbulent diffusion
shearing
statistics
Renormalization
Velocity Field
Scalar
Statistics
Order Statistics
Random Field
velocity distribution
shear layers
Divergence
scalars
Infrared
Phase Transition
Range of data
divergence
Advection-diffusion
Temporal Correlation

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

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

In: Journal of Statistical Physics, Vol. 75, No. 5-6, 06.1994, p. 1153-1165.

Research output: Contribution to journalArticle

Majda, Andrew J. / Random shearing direction models for isotropic turbulent diffusion. In: Journal of Statistical Physics. 1994 ; Vol. 75, No. 5-6. pp. 1153-1165.
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