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Mixing by Statistically Self-similar Gaussian Random Fields

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Abstract

We study the passive transport of a scalar field by a spatially smooth but white-in-time incompressible Gaussian random velocity field on \(\mathbb {R}^d\). 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 \(\mathbb {E}\ \Vert \theta _t \Vert _{\dot{H}^{-s}}^2 = \textrm{e}^{-\lambda _{d,s} t} \Vert \theta _0 \Vert _{\dot{H}^{-s}}^2\) with any \(s\in (0,d/2)\) and \(\frac{\lambda _{d,s}}{D_1}:= s(\frac{\lambda _{1}}{D_1}-2s)\) where \(\lambda _1/D_1 = d\) is the top Lyapunov exponent associated to the random Lagrangian flow generated by u and \( D_1\) is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold uniformly in diffusivity.

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Notes

  1. The physical dimensions of \(D_1\) are inverse time and it can be regarded as a proxy for the shear rate at small scales. See the discussion of Kraichnan, e.g. [23, Eq. (3.5)] This shows that \(D_1\) is essentially a square of the fine-scale turbulent shear-rate times a Lagrangian correlation time of the shear rate. We note that our notation differs slightly different from that used in recent literature which replaces \(D_1\rightarrow (d-1) D_1\).

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Acknowledgements

We thank G. Eyink, A. Frishman and S. Punshon-Smith and the anonymous referee for useful comments. The research of MCZ was partially supported by the Royal Society URF\(\backslash \)R1\(\backslash \)191492 and EPSRC Horizon Europe Guarantee EP/X020886/1. The research of TDD was partially supported by the NSF DMS-2106233 grant and NSF CAREER award #2235395. The research of RSG was partially supported by the Deutsche Forschungsgemeinschaft through the SPP 2410/1 Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness. RSG’s research took place within the scope of the NCCR SwissMAP which was funded by the Swiss National Science Foundation (grant number 205607).

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Michele Coti Zelati

  2. Department of Mathematics, Stony Brook University, 11790, Stony Brook, NY, USA

    Theodore D. Drivas

  3. Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Rishabh S. Gvalani

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  1. Michele Coti Zelati
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Correspondence to Theodore D. Drivas.

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Coti Zelati, M., Drivas, T.D. & Gvalani, R.S. Mixing by Statistically Self-similar Gaussian Random Fields. J Stat Phys 191, 61 (2024). https://doi.org/10.1007/s10955-024-03277-w

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