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Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows

Dallas Albritton, Rajendra Beekie

TL;DR

This work proves sharp, uniform-in-diffusivity mixing rates for passive scalars advected by parallel shear flows, establishing $\| f(t) \|_{H^{-1}} \lesssim \langle t \rangle^{-1/(N+1)}$ where $N$ is the maximal order of vanishing of $b'(y)$ at critical points. It develops a resolvent-based framework, with precise Airy-type kernel bounds, to decompose the dynamics into inner boundary-layer regions near critical points and outer regions, enabling a global understanding of mixing and enhanced dissipation. In the non-degenerate case $N=1$, it provides a rigorous asymptotic description of generic solutions as decaying traveling waves localized to shear layers of width $\kappa^{1/4}$, validating predictions in the physics literature and revealing an explicit spectral mechanism for long-time behavior. The results unify and extend prior monotone/shear-flow analyses, clarify the role of critical points in uniform-in-$\kappa$ mixing, and deliver high-order spectral asymptotics for the slowest eigenmodes, with implications for Batchelor-scale-like length scales and long-time transport in shear-dominated flows.

Abstract

We consider the advection-diffusion equation describing the evolution of a passive scalar in a background shear flow. We prove the optimal uniform-in-diffusivity mixing rate $\| f \|_{H^{-1}} \lesssim \langle t \rangle^{-1/(N+1)}$, $t \geq 0$, where $N$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile, e.g., $N=1$ for plane Pouseille flow. Our proof is based on the description of the solution in terms of resolvents and involves pointwise estimates on the resolvent kernel. In the non-degenerate case, we further give a rigorous asymptotic description of generic solutions in terms of shear layers localized around the critical points. This verifies formal asymptotics in [McLaughlin-Camassa-Viotti, \textit{Physics of Fluids}, 22(11), 2010].

Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows

TL;DR

This work proves sharp, uniform-in-diffusivity mixing rates for passive scalars advected by parallel shear flows, establishing where is the maximal order of vanishing of at critical points. It develops a resolvent-based framework, with precise Airy-type kernel bounds, to decompose the dynamics into inner boundary-layer regions near critical points and outer regions, enabling a global understanding of mixing and enhanced dissipation. In the non-degenerate case , it provides a rigorous asymptotic description of generic solutions as decaying traveling waves localized to shear layers of width , validating predictions in the physics literature and revealing an explicit spectral mechanism for long-time behavior. The results unify and extend prior monotone/shear-flow analyses, clarify the role of critical points in uniform-in- mixing, and deliver high-order spectral asymptotics for the slowest eigenmodes, with implications for Batchelor-scale-like length scales and long-time transport in shear-dominated flows.

Abstract

We consider the advection-diffusion equation describing the evolution of a passive scalar in a background shear flow. We prove the optimal uniform-in-diffusivity mixing rate , , where is the maximal order of vanishing of the derivative of the shear profile, e.g., for plane Pouseille flow. Our proof is based on the description of the solution in terms of resolvents and involves pointwise estimates on the resolvent kernel. In the non-degenerate case, we further give a rigorous asymptotic description of generic solutions in terms of shear layers localized around the critical points. This verifies formal asymptotics in [McLaughlin-Camassa-Viotti, \textit{Physics of Fluids}, 22(11), 2010].

Paper Structure

This paper contains 23 sections, 19 theorems, 362 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Optimal mixing rates
  3. Asymptotic description and limiting length scale
  4. Comparison with existing literature
  5. Dimensional analysis
  6. Mixing via resolvents
  7. Airy bounds
  8. Overview of the proof
  9. Proof of Theorem \ref{['thm:1']}
  10. Changes needed for $j = 0$
  11. Comparison with J23BCJ24
  12. Slowest eigenmodes and asymptotic length scale
  13. Gluing framework
  14. The inner equation
  15. Outer problem solvability
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $b : \mathbb{T} \to \mathbb{R}$ satisfy Assumption assump:b and $f^{\rm in} \in H^1(\mathbb{T})$ with zero mean on streamlines: There exist $c, \kappa_0 \in (0,1]$, depending only on $b$, such that for $\kappa \in (0,\kappa_0]$, the solution $f : [0,+\infty) \times \mathbb{T}^2 \to \mathbb{R}$ to adv:diff with initial datum $f^{\rm in}$ satisfies the uniform-in-diffusivity mixing and enhanced

Figures (4)

  • Figure 1.1: Left: First 25, 12, and 8 eigenvalues of the operator $L_\varepsilon = \varepsilon \partial_y^2 - i \sin y$ with sinusoidal shear flow and $\varepsilon = 0.01, 0.05, 0.1$, respectively. Compare TrefethenEmbree for the complex Airy operator $\varepsilon \partial_y^2 + iy$ with Dirichlet boundary conditions. Right: First 4 normalized eigenvalues $\Lambda_\alpha$, $\alpha = 0, 1, 2, 3$, in the notation of \ref{['eq:lambdascaling']} and \ref{['eq:lambdascaling2']}, from the "lower branch": $\gamma = \pi/2$ and $b(\gamma) = 1$. Here, $\tau = (2/\varepsilon)^{1/2}$ is the scaling factor in \ref{['eq:lambdascaling']}.
  • Figure 1.2: Bottom eigenfunction, concentrated around $y = \pi/2$, of the operator $L_\varepsilon = \varepsilon \partial_y^2 - i \sin y$ with sinusoidal shear flow and $\varepsilon = 10^{-3}, 10^{-2}, 10^{-1}$.
  • Figure 1.3: Heat plots at time $T$ of a passive scalar with initial datum $f^{\rm in} = \exp(-|x-\pi|^2) - \langle \exp(-|x-\pi|^2) \rangle$ evolving in a sinusoidal shear flow $b(y) = \sin y$. Left: Diffusivity $\kappa=0.08$, $T=22$. Right: Diffusivity $\kappa=0.005$, $T=32$. Compare Camassa2007Camassa2010.
  • Figure 3.1: Cut-off functions $\chi_O$, $\chi_I$ with $\ell=1$ demonstrating overlap and shear profile $b(y) = (y-\gamma_1)^2/2$.

Theorems & Definitions (53)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 43 more