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Article

Caustic Frequency in 2D Stochastic Flows Modeling Turbulence

by
Leonid I. Piterbarg
Department of Mathematics, University of Southern California, Kaprielian Hall, Room 108, 3620 Vermont Avenue, Los Angeles, CA 90089-2532, USA
Mathematics 2021, 9(8), 797; https://doi.org/10.3390/math9080797
Submission received: 8 February 2021 / Revised: 21 March 2021 / Accepted: 26 March 2021 / Published: 7 April 2021
(This article belongs to the Special Issue Stochastic Processes and Random Fields)

Abstract

Stochastic flows mimicking 2D turbulence in compressible media are considered. Particles driven by such flows can collide and we study the collision (caustic) frequency. Caustics occur when the Jacobian of a flow vanishes. First, a system of nonlinear stochastic differential equations involving the Jacobian is derived and reduced to a smaller number of unknowns. Then, for special cases of the stochastic forcing, upper and lower bounds are found for the mean number of caustics as a function of Stokes number. The bounds yield an exact asymptotic for small Stokes numbers. The efficiency of the bounds is verified numerically. As auxiliary results we give rigorous proofs of the well known expressions for the caustic frequency and Lyapunov exponent in the one-dimensional model. Our findings may also be used for estimating the mean time when a 2D Riemann type partial differential equation with a stochastic forcing loses uniqueness of solutions.
Keywords: stochastic flows; inertial particles; caustics; Lagrangian stochastic models stochastic flows; inertial particles; caustics; Lagrangian stochastic models

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MDPI and ACS Style

Piterbarg, L.I. Caustic Frequency in 2D Stochastic Flows Modeling Turbulence. Mathematics 2021, 9, 797. https://doi.org/10.3390/math9080797

AMA Style

Piterbarg LI. Caustic Frequency in 2D Stochastic Flows Modeling Turbulence. Mathematics. 2021; 9(8):797. https://doi.org/10.3390/math9080797

Chicago/Turabian Style

Piterbarg, Leonid I. 2021. "Caustic Frequency in 2D Stochastic Flows Modeling Turbulence" Mathematics 9, no. 8: 797. https://doi.org/10.3390/math9080797

APA Style

Piterbarg, L. I. (2021). Caustic Frequency in 2D Stochastic Flows Modeling Turbulence. Mathematics, 9(8), 797. https://doi.org/10.3390/math9080797

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