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The Obukhov--Corrsin spectrum of passive scalar turbulence through anomalous regularization

Keefer Rowan

TL;DR

This work establishes a rigorous Obukhov–Corrsin-type spectral scaling for a passive scalar advected by rough, turbulent-like velocity fields, showing that the spectral mass on Fourier annuli follows the predicted $|k|^{-d- (1-\alpha)}$ law when summed over annuli, up to logarithmic factors. The main tool is a sharp anomalous regularization result, proven for Kraichnan-type models via a Fourier-space $\ell^p$ energy equality and a weighted lattice Poincaré inequality, with uniform-in-diffusivity control. The authors prove anomalous dissipation and regularization for two Kraichnan models (isotropic and shear) and extend the OC lower bounds to correlated-in-time and white-in-time velocity fields, thereby providing a first complete dynamical justification of OC-type scaling in rough-velocity regimes. The results bridge the Batchelor and OC paradigms, offering precise Fourier-space localization and illuminating the interplay between spectral smoothing, dissipation, and turbulent forcing in stochastic advection-diffusion systems.

Abstract

The Obukhov--Corrsin spectrum predicts the distribution of Fourier mass for a passive scalar field advected by a "turbulent" velocity field with spatial regularity $C^α_x$ for $α\in (0,1)$ and subject to a time-stationary forcing. We prove the Obukhov--Corrsin spectrum holds after summing over geometric annuli in Fourier space -- up to logarithmic corrections -- as a consequence of a sharp anomalous regularization result. We then prove this anomalous regularization for a broad class of Kraichnan-type models. The proof of anomalous regularization relies on a Fourier space $\ell^p$ energy equality and a weighted lattice Poincaré inequality.

The Obukhov--Corrsin spectrum of passive scalar turbulence through anomalous regularization

TL;DR

This work establishes a rigorous Obukhov–Corrsin-type spectral scaling for a passive scalar advected by rough, turbulent-like velocity fields, showing that the spectral mass on Fourier annuli follows the predicted law when summed over annuli, up to logarithmic factors. The main tool is a sharp anomalous regularization result, proven for Kraichnan-type models via a Fourier-space energy equality and a weighted lattice Poincaré inequality, with uniform-in-diffusivity control. The authors prove anomalous dissipation and regularization for two Kraichnan models (isotropic and shear) and extend the OC lower bounds to correlated-in-time and white-in-time velocity fields, thereby providing a first complete dynamical justification of OC-type scaling in rough-velocity regimes. The results bridge the Batchelor and OC paradigms, offering precise Fourier-space localization and illuminating the interplay between spectral smoothing, dissipation, and turbulent forcing in stochastic advection-diffusion systems.

Abstract

The Obukhov--Corrsin spectrum predicts the distribution of Fourier mass for a passive scalar field advected by a "turbulent" velocity field with spatial regularity for and subject to a time-stationary forcing. We prove the Obukhov--Corrsin spectrum holds after summing over geometric annuli in Fourier space -- up to logarithmic corrections -- as a consequence of a sharp anomalous regularization result. We then prove this anomalous regularization for a broad class of Kraichnan-type models. The proof of anomalous regularization relies on a Fourier space energy equality and a weighted lattice Poincaré inequality.

Paper Structure

This paper contains 22 sections, 19 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Results for two Kraichnan models
  3. Notation and general results
  4. Notation, definitions, and assumptions
  5. Bounds on the energy spectrum of the invariant measure
  6. Anomalous regularization and dissipation for white-in-time models
  7. Discussion and previous results
  8. Anomalous dissipation, the Obukhov--Corrsin spectrum, and the importance of anomalous regularization
  9. The Kraichnan model/transport noise
  10. Anomalous regularization
  11. Batchelor regime passive scalars
  12. Uniform-in-diffusivity vs. at zero diffusivity
  13. Overview of the argument
  14. Overview of the proof of anomalous regularization and anomalous dissipation in white-in-time models
  15. Overview of the proof of the Obukhov--Corrsin lower bounds
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $d\geq 2$, $\alpha \in (0,1)$, and let $du_t$ have its coefficients defined by either Definition defn:Kraichnan standard or Definition defn:Kraichnan shear. Then there exists $C(d,\alpha)>0$ such that for all $F \in L^2(\mathbb{T}^d)$ with $\int F(x)\,dx = 0$ and all $\kappa>0$, if we let $\varp where $\widehat{\varphi}^\kappa_t(k)$ is the $k$th Fourier coefficient of $\varphi^\kappa_t$.

Theorems & Definitions (42)

  • Definition 1.1: Isotropic Kraichnan model on $\mathbb{T}^d$
  • Definition 1.2: Shear Kraichnan model on $\mathbb{T}^d$
  • Theorem 1.3: Anomalous dissipation and regularization
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Obukhov--Corrsin spectrum on annuli
  • Corollary 1.7: Infinite order smoothing in constant directions
  • Definition 1.8: $\sigma(H)$-spaces
  • Definition 1.9: Anomalous regularization and dissipation
  • Proposition 1.10: Invariant measures
  • ...and 32 more