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Homogenisation of a Passive Scalar Transported by Locally Supported White Noise

Federico Butori, Avi Mayorcas, Silvia Morlacchi

TL;DR

This work analyzes the scaling limit of a stochastic passive scalar equation driven by anisotropic, locally supported white-noise transport, showing that as the patch centers become dense ($N\to\infty$) the SPDE converges to a deterministic diffusion equation with an effective diffusivity $C(c,\kappa)$. The authors combine martingale control and homogenization of the Stratonovich corrector to identify a limit PDE of the form $\partial_t \bar u = C(c,\kappa)\Delta\bar u$, and provide a detailed variational and regime-dependent analysis of $C(c,\kappa)$. They prove that the homogenized diffusivity is diagonal and characterize how it depends on the overlap parameter $c$ and the molecular diffusivity $\kappa$, with distinct behaviors in sparse, intermediate, and highly overlapped regimes; in particular, for $c>\sqrt{5}/2$ the noise induces uniform ellipticity, while for $\sqrt{2}/2\le c\le \sqrt{5}/2$ there can be genuine degeneracy yet a positive lower bound persists. Numerical simulations corroborate the theoretical findings and reveal nontrivial $\kappa\to 0$ asymptotics, including nonlinear growth in certain regimes and a transition in the nature of the intercept depending on $c$. Collectively, the results provide a rigorous link between Itô-Stratonovich diffusion limits and homogenization, with implications for modeling stratified turbulence and boundary-layer transport.

Abstract

Stochastic perturbations of transport type are a common and widely accepted way of representing turbulent effects in fluid dynamics models. In many known examples, it even leads to improved solution theory, a phenomenon known as \emph{regularization by noise}. A common thread in the recent literature on the topic is the so-called \emph{Itô-Stratonovich diffusion limit}. By selecting Stratonovich transport noise with carefully arranged vector fields, one can show that the solution of certain SPDEs are close, in an appropriate topology, to an effective, deterministic, equation with a new effective second order elliptic operator, linked to the Ito-Stratonovich corrector. In this work, we deal with a passive scalar model with molecular diffusivity $κ$. Starting from the results in [Flandoli \emph{et al.}, 2022, \emph{Philos. Trans. Roy. Soc. A}, 380(2219)], we consider a transport noise made by a sum of independent and compactly supported vector fields. This setting is relevant for models of stratified turbulence which naturally occur in boundary layers and Boussinesq models. Due to the anisotropic nature of the noise, the identification of the limit equation is not straightforward as in all other examples known in literature, as the Ito-Stratonovich corrector is a generic second order elliptic operator with non-constant coefficients. Using tools from Homogenisation theory, we obtain a representation for the limiting effective diffusivity matrix. Exploiting this representation, we study asymptotics, in the $κ\rightarrow 0$ regime, of the effective diffusivity across a number of vector field regimes parametrised by the radius of their support. Finally, we provide a careful numerical analysis of the effective diffusivity, discovering a nonlinear behavior for $κ\rightarrow 0$, in some regimes.

Homogenisation of a Passive Scalar Transported by Locally Supported White Noise

TL;DR

This work analyzes the scaling limit of a stochastic passive scalar equation driven by anisotropic, locally supported white-noise transport, showing that as the patch centers become dense () the SPDE converges to a deterministic diffusion equation with an effective diffusivity . The authors combine martingale control and homogenization of the Stratonovich corrector to identify a limit PDE of the form , and provide a detailed variational and regime-dependent analysis of . They prove that the homogenized diffusivity is diagonal and characterize how it depends on the overlap parameter and the molecular diffusivity , with distinct behaviors in sparse, intermediate, and highly overlapped regimes; in particular, for the noise induces uniform ellipticity, while for there can be genuine degeneracy yet a positive lower bound persists. Numerical simulations corroborate the theoretical findings and reveal nontrivial asymptotics, including nonlinear growth in certain regimes and a transition in the nature of the intercept depending on . Collectively, the results provide a rigorous link between Itô-Stratonovich diffusion limits and homogenization, with implications for modeling stratified turbulence and boundary-layer transport.

Abstract

Stochastic perturbations of transport type are a common and widely accepted way of representing turbulent effects in fluid dynamics models. In many known examples, it even leads to improved solution theory, a phenomenon known as \emph{regularization by noise}. A common thread in the recent literature on the topic is the so-called \emph{Itô-Stratonovich diffusion limit}. By selecting Stratonovich transport noise with carefully arranged vector fields, one can show that the solution of certain SPDEs are close, in an appropriate topology, to an effective, deterministic, equation with a new effective second order elliptic operator, linked to the Ito-Stratonovich corrector. In this work, we deal with a passive scalar model with molecular diffusivity . Starting from the results in [Flandoli \emph{et al.}, 2022, \emph{Philos. Trans. Roy. Soc. A}, 380(2219)], we consider a transport noise made by a sum of independent and compactly supported vector fields. This setting is relevant for models of stratified turbulence which naturally occur in boundary layers and Boussinesq models. Due to the anisotropic nature of the noise, the identification of the limit equation is not straightforward as in all other examples known in literature, as the Ito-Stratonovich corrector is a generic second order elliptic operator with non-constant coefficients. Using tools from Homogenisation theory, we obtain a representation for the limiting effective diffusivity matrix. Exploiting this representation, we study asymptotics, in the regime, of the effective diffusivity across a number of vector field regimes parametrised by the radius of their support. Finally, we provide a careful numerical analysis of the effective diffusivity, discovering a nonlinear behavior for , in some regimes.

Paper Structure

This paper contains 28 sections, 25 theorems, 166 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Elements of Homogenization Theory
  4. Setting
  5. Definition of the Noise
  6. Solution Concepts and Basic Properties
  7. Discussion of the Scaling Limit
  8. Main Results
  9. The Scaling Limit
  10. Controlling the Martingale
  11. Homogenization of the Stratonovich Corrector
  12. Combined Homogenised and Itô-Stratonovich Diffusion Limit
  13. Analysis of Homogenised Diffusivity
  14. Properties of the covariance Matrices
  15. Variational Characterisation of the Homogenised Operator
  16. ...and 13 more sections

Key Result

Lemma 1.1

Let $M(x)$ be a smooth and periodic field of symmetric matrices and let $0\leq \lambda \leq \Lambda <+\infty$ be such that Furthermore, we fix $\phi_i\in \mathcal{H}^1(\mathbb{T}^2)$, for $i=1,2$, weak solutions to eq:corrector_gen. Then, the following facts hold.

Figures (7)

  • Figure 1: Illustration of the supports of $A(x)$ for selected $c \in \{0.25,\, 0.5,\, 0.75,\, 1,\, 1.25,\, 1.5\}$. Darker shading represent multiple overlaps.
  • Figure 2: Up: plot of $\varphi(x) / \|\varphi\|_{L^2_x}$, with $\varphi(x)$ defined in \ref{['eq:phi_numerics']}, with our choice $a_1 = 0.05$, $a_2 = 0.3$, at $c=1.2$. This choice of the parameter $c$ shows that for $c>1$ the profile function is not zero outside the interval [-1,1]. Down: different possible choices of the coefficients $a_1$ (left, $a_2$ is held fixed, $a_2 = 0.3$), $a_2$ (right, $a_1$ is held fixed, $a_1 = 0.05$), at $c=1.2$. Choices of parameters employed in numerical simulations are plotted in blue, with $a_1 = 0.05$, $a_2 = 0.3$. Note that the plot on the right hints that for $a_2 > 0.6$ the support of the function drops to a value compatible with zero already before $c=1.2$.
  • Figure 3: Plot of additional diffusivity $C(c, \kappa)$, for different values of $c$, from left to right and top to bottom: $c\in (0, 1/2]$,$c\in(1/2, \sqrt{2}/2)$, $c \in (\sqrt{2}/2, \sqrt{5}/2)$, $c> \sqrt{5}/2$ (with zoom at $\kappa \to 0$ if needed). The additional diffusivity is computed by means of \ref{['eq:total_viscosity']}. Compatibility with results obtained by using \ref{['eq:total_viscosity_var']} was separately checked. The profile function used is $\varphi(x)$, defined in \ref{['eq:phi_numerics']}, with $a_1 = 0.05$, $a_2 = 0.3$. The top right panel exhibits values of $c$ in the range $c\in (1/2, \sqrt{2}/2)$, which we highlight lies outside the statements of \ref{['th:main_introduction']}.
  • Figure 4: Intercept estimate (left) and power law exponent estimate (right) as a function of the parameter $c$. The profile function used is $\varphi(x)$, defined in \ref{['eq:phi_numerics']}, with $a_1 = 0.05$, $a_2 = 0.3$.
  • Figure 5: Plot of additional diffusivity, for the values of the parameter $c = 0.4$ (up, left), $c = 0.6$ (up, right), $c = 1.0$ (down, left), $c = 1.4$ (down, right), computed as in \ref{['eq:total_viscosity']}. Compatibility with results obtained by using \ref{['eq:total_viscosity_var']} was separately checked. The profile function used is $\varphi(x)$, defined in \ref{['eq:phi_numerics']}, with $a_1 = 0.05, 0.5, 1.0$, $a_2 = 0.3$, and $a_1 = 0.05$, $a_2 = 0.08, 0.3, 0.6$.
  • ...and 2 more figures

Theorems & Definitions (62)

  • Lemma 1.1
  • proof
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3: Mean Free Test Functions
  • Proposition 2.4: Energy Estimate
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • ...and 52 more