Statistically self-similar mixing by Gaussian random fields

2023/09/27 by Zelati, Michele Coti, Drivas, Theodore D., Gvalani, Rishabh S. · 1 citation
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2309.15744

Abstract

We study the passive transport of a scalar field by a spatially smooth but white-in-time incompressible Gaussian random velocity field on ℝ^d. If the velocity field u is homogeneous, isotropic, and statistically self-similar, we derive an exact formula which captures non-diffusive mixing. For zero diffusivity, the formula takes the shape of 𝔼 ‖ θ_t ‖_H^-s2 = e^-λ_d,s t ‖ θ₀ ‖_H^-s2 with any s∈ (0,d/2) and \fracλ_d,sD₁:= s(\fracλ₁D₁-2s) where λ₁/D₁ = d is the top Lyapunov exponent associated to the random Lagrangian flow generated by u and D₁ is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold uniformly in diffusivity.

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• Mixing at the Batchelor Scale for White-In-Time Flows

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