TY - JOUR

T1 - An explicit family of probability measures for passive scalar diffusion in a random flow

AU - Bronski, Jared C.

AU - Camassa, Roberto

AU - Lin, Zhi

AU - McLaughlin, Richard M.

AU - Scotti, Alberto

N1 - Funding Information:
The authors thank Ken McLaughlin for helpful discussions. RMM, ZL, and AS were partially supported by a National Science Foundation Collaborations in Mathematical Geosciences (CMG) Award, NSF ATM-0327906. RMM was also partially supported by NSF DMS-030868. RC was partially supported by NSF DMS-0104329 and DMS-0509423. JCB was partially supported by NSF DMS-0354462. Computational work was supported by DMS-SCREMS 0422417. All wish to thank the suggestion made by the reviewers, especially regarding the connection between some of the singularities in the PDF and the extrema in the initial scalar field, and to explore the role of the scalar source in this model.

PY - 2007/8

Y1 - 2007/8

N2 - 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).

AB - 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).

KW - Monte-Carlo simulations

KW - Orthogonal polynomials

KW - Probability measures

KW - Turbulent transport

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U2 - 10.1007/s10955-007-9316-y

DO - 10.1007/s10955-007-9316-y

M3 - Article

AN - SCOPUS:36148947876

VL - 128

SP - 927

EP - 968

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -