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

Robin Chemnitz, Dennis Chemnitz

TL;DR

The paper proves a uniform-in-diffusivity lower bound on the exponential dissipation rate for a white-in-time 2D (and 3D) four-mode passive scalar model, thereby verifying Batchelor-scale behavior (filamentation length scaling like $\sqrt{\kappa}$) when diffusion is small. It complements existing upper bounds to establish that the Batchelor-scale conjecture holds in this tractable stochastic advection-diffusion setting, and it provides a detailed analysis of mixing rates at zero diffusion, showing $\gamma_s \sim 1\wedge s$. The core method combines Fourier-space SDEs with a drift-positivity argument, stopping times, and a careful independence-from-initial-condition analysis, and the approach extends naturally to a three-dimensional, 12-mode analogue. This work advances rigorous understanding of passive-scalar turbulence, offering a concrete model where Batchelor-scale predictions are verified and shedding light on the role of white-in-time velocity fields in mixing dynamics.

Abstract

We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from below. Together with the corresponding upper bound established by Gess and Yaroslavtsev, this constitutes an example of a velocity field for which the Batchelor scale conjecture can be verified. In addition, we characterize the exponential mixing rate without diffusion of this system. Our results are not restricted to two dimensions, and we construct a three-dimensional white-in-time velocity field with the same properties.

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

TL;DR

The paper proves a uniform-in-diffusivity lower bound on the exponential dissipation rate for a white-in-time 2D (and 3D) four-mode passive scalar model, thereby verifying Batchelor-scale behavior (filamentation length scaling like ) when diffusion is small. It complements existing upper bounds to establish that the Batchelor-scale conjecture holds in this tractable stochastic advection-diffusion setting, and it provides a detailed analysis of mixing rates at zero diffusion, showing . The core method combines Fourier-space SDEs with a drift-positivity argument, stopping times, and a careful independence-from-initial-condition analysis, and the approach extends naturally to a three-dimensional, 12-mode analogue. This work advances rigorous understanding of passive-scalar turbulence, offering a concrete model where Batchelor-scale predictions are verified and shedding light on the role of white-in-time velocity fields in mixing dynamics.

Abstract

We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from below. Together with the corresponding upper bound established by Gess and Yaroslavtsev, this constitutes an example of a velocity field for which the Batchelor scale conjecture can be verified. In addition, we characterize the exponential mixing rate without diffusion of this system. Our results are not restricted to two dimensions, and we construct a three-dimensional white-in-time velocity field with the same properties.

Paper Structure

This paper contains 12 sections, 10 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dissipation of a passive scalar in the Batchelor regime
  3. Almost-sure exponential mixing in the Batchelor regime
  4. White-in-time velocity fields and main results
  5. Main Results
  6. Three-dimensional model
  7. Related literature
  8. Proof of the Main Result
  9. Proof of the Lower Bound
  10. Independence of the initial condition
  11. Three-dimensional model
  12. Well posedness of the transport SPDE and estimates on the mixing rate $\gamma_s$

Key Result

Theorem 2.1

For $\kappa\geq 0$ and $f_0\in L_0^2(\mathds{T}^2) \setminus \{0\}$, let $f_t$ be the solution to spde:main2D. The decay rate of the $L^2$-norm of the four innermost modes is almost-surely lower bounded by

Figures (2)

  • Figure 1: Each column represents a typical realization of the normalized solution $\pi_t \coloneqq f_t / \left\lVert f_t\right\rVert_{L^2}$ to the 4-modes model \ref{['spde:main2D']} for different values of $\kappa$; from left to right $\kappa=0.04, \, 0.01,\, 0.0025$. The first row depicts the function $\pi_t$ on the torus, and the predicted filamentation length $\sqrt{\kappa}$ is marked in red. The second row shows the Fourier spectrum of $\pi_t$ as a heatmap on $\mathds{Z}^2$. A multiple of the Batchelor scale ($10\,\kappa^{-\frac{1}{2}}$) is drawn as a red circle. The third row depicts the power spectrum of $\pi_t$ with a red line at $10\, \kappa^{-\frac{1}{2}}$.
  • Figure 2: Fourier modes $\hat{f}_{k,l}(t)$ of the solution $f_t$ to \ref{['spde:advDiff']}. The inner modes with $k^2+l^2= 1$ are drawn as solid dots while all other modes are drawn as hollow dots. The arrows indicate which modes interact via the transport term; see Proposition \ref{['prop:Fourier_representation']}.

Theorems & Definitions (17)

  • Conjecture 1: Batchelor Scale Conjecture
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:lower_bound_low_modes']}
  • ...and 7 more