Non-Gaussian invariant measures for the majda model of decaying turbulent transport

Research output: Contribution to journalArticle

Abstract

The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large-deviation problem for the scalar field into a small-deviation, or small-ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here such a methodology is applied to a model proposed by Majda consisting of a white-in-time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non-Gaussian invariant measure for the (reduced) scalar field is derived, and, in particular, it is shown that the one-point distribution of the scalar has stretched exponential tails, with a stretching exponent depending on the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one-to-one correspondence with an exponent measuring time persistence effects in the velocity field.

Original languageEnglish (US)
Pages (from-to)1146-1167
Number of pages22
JournalCommunications on Pure and Applied Mathematics
Volume54
Issue number9
DOIs
StatePublished - Sep 2001

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Invariant Measure
Scalar Field
Velocity Field
Scalar
Tail
Exponent
Shear flow
Fluctuations
Oils and fats
Probability distributions
Small Deviations
Path
Stretching
Shear Flow
One to one correspondence
Large Deviations
Model
Persistence
Random Field
Universality

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Non-Gaussian invariant measures for the majda model of decaying turbulent transport. / Vanden Eijnden, Eric.

In: Communications on Pure and Applied Mathematics, Vol. 54, No. 9, 09.2001, p. 1146-1167.

Research output: Contribution to journalArticle

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