Elementary models with probability distribution function intermittency for passive scalars with a mean gradient

A. Bourlioux, A. J. Majda

Research output: Contribution to journalArticlepeer-review


The single-point probability distribution function (PDF) for a passive scalar with an imposed mean gradient is studied here. Elementary models are introduced involving advection diffusion of a passive scalar by a velocity field consisting of a deterministic or random shear flow with a transverse time-periodic transverse sweep. Despite the simplicity of these models, the PDFs exhibit scalar intermittency, i.e., a transition from a Gaussian PDF to a broader than Gaussian PDF with large variance as the Péclet number increases with a universal self-similar shape that is determined analytically by explicit formulas. The intermittent PDFs resemble those that have been found recently in numerical simulations of much more complex models. The examples presented here unambiguously demonstrate that neither velocity fields inducing chaotic particle trajectories with positive Lyapunov exponents nor strongly turbulent velocity fields are needed to produce scalar intermittency with an imposed mean gradient. The passive scalar PDFs in these models are given through exact solutions that are processed in a transparent fashion via elementary stationary phase asymptotics and numerical quadrature of one-dimensional formulas.

Original languageEnglish (US)
Pages (from-to)881-897
Number of pages17
JournalPhysics of Fluids
Issue number2
StatePublished - Feb 2002

ASJC Scopus subject areas

  • Computational Mechanics
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Fluid Flow and Transfer Processes


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