An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow |
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Authors: | Jared C Bronski Roberto Camassa Zhi Lin Richard M McLaughlin Alberto Scotti |
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Institution: | (1) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA;(2) Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA;(3) Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA;(4) CB#3250, Phillips Hall, University of North Carolina, Chapel Hill, NC 27599-3250, USA |
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Abstract: | We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing
in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive
an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for
validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion.
With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully
benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study
of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently
tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar
PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from
an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Péclet number is shown to
be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at
unit scalar values (small Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet). |
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Keywords: | Turbulent transport probability measures orthogonal polynomials Monte-Carlo simulations |
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