Random shearing direction models for isotropic turbulent diffusion

Andrew J. Majda

Research output: Contribution to journalArticlepeer-review


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
Issue number5-6
StatePublished - Jun 1994


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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