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Transport in multifractal Kraichnan flows: from turbulence to Liouville quantum gravity

André L. P. Considera, Simon Thalabard

TL;DR

The paper addresses how spatial intermittency in turbulent transport modifies Lagrangian dispersion by introducing a multifractal Kraichnan model that couples a 1D Kraichnan velocity to a frozen Gaussian multiplicative chaos. It develops a rigorous, diffusion-based analysis using 1D Feller theory to classify phase transitions of the two-particle separation under quenched and annealed randomness, revealing a smoothing-by-intermittency effect. A key result is that the quenched phase structure is governed by the most probable Hölder exponent of the velocity field, while annealing produces additional smoothing and can be mapped to mean-field exponents; the analysis also links the MK process to a one-dimensional multiplicative Liouville Brownian motion (MLBM) and shows a precise parameter mapping between MK and MLBM. This work unifies turbulence intermittency with Liouville quantum gravity ideas, showing how a random-GMC geometry shapes diffusion and leads to a coherent phase diagram, with potential extensions to higher dimensions and multiparticle dynamics.

Abstract

We investigate the behavior of fluid trajectories in a multifractal extension of the Kraichnan model of turbulent advection. The model couples a one-dimensional, Gaussian, white-in-time random flow to a frozen-in-time Gaussian multiplicative chaos (GMC). The resulting velocity field features an interplay between the roughness exponent $ξ\in(0,2]$, controlling the correlation decay for the Gaussian component, and the intermittency parameter $γ\in [0,\sqrt {2}/2)$, prescribing the deviations from self-similarity. Recent numerical work by the authors suggests that such coupling induce a smoothing-by-intermittency effect, and the purpose here is to address this phenomenon theoretically. Using the theory of 1D Feller Markov processes, we characterize the phases of the two-particle separation process upon varying $ξ$ and $γ$, extending to a multifractal setting the stochastic/deterministic and colliding/non-colliding transitions known in the monofractal Kraichnan case. Our analysis distinguishes between two settings: quenched or annealed. In the quenched setting, the GMC realization is prescribed, and we show that the phases are governed by the most probable Hölder exponent of the multifractal velocity field. In the annealed setting, the GMC is averaged over, leading to an additional smoothing effect. Moreover, we show that the separation process exhibits structural analogies with multiplicative one-dimensional versions of the Liouville Brownian motion (LBM)--a diffusion process evolving in a random GMC landscape, originally introduced in the context of Liouville quantum gravity. In particular, both the quenched and annealed phase transitions are recovered by considering a multiplicative LBM characterized by a roughness parameter $ξ+ 4γ^2$ and an intermittency exponent $γ$.

Transport in multifractal Kraichnan flows: from turbulence to Liouville quantum gravity

TL;DR

The paper addresses how spatial intermittency in turbulent transport modifies Lagrangian dispersion by introducing a multifractal Kraichnan model that couples a 1D Kraichnan velocity to a frozen Gaussian multiplicative chaos. It develops a rigorous, diffusion-based analysis using 1D Feller theory to classify phase transitions of the two-particle separation under quenched and annealed randomness, revealing a smoothing-by-intermittency effect. A key result is that the quenched phase structure is governed by the most probable Hölder exponent of the velocity field, while annealing produces additional smoothing and can be mapped to mean-field exponents; the analysis also links the MK process to a one-dimensional multiplicative Liouville Brownian motion (MLBM) and shows a precise parameter mapping between MK and MLBM. This work unifies turbulence intermittency with Liouville quantum gravity ideas, showing how a random-GMC geometry shapes diffusion and leads to a coherent phase diagram, with potential extensions to higher dimensions and multiparticle dynamics.

Abstract

We investigate the behavior of fluid trajectories in a multifractal extension of the Kraichnan model of turbulent advection. The model couples a one-dimensional, Gaussian, white-in-time random flow to a frozen-in-time Gaussian multiplicative chaos (GMC). The resulting velocity field features an interplay between the roughness exponent , controlling the correlation decay for the Gaussian component, and the intermittency parameter , prescribing the deviations from self-similarity. Recent numerical work by the authors suggests that such coupling induce a smoothing-by-intermittency effect, and the purpose here is to address this phenomenon theoretically. Using the theory of 1D Feller Markov processes, we characterize the phases of the two-particle separation process upon varying and , extending to a multifractal setting the stochastic/deterministic and colliding/non-colliding transitions known in the monofractal Kraichnan case. Our analysis distinguishes between two settings: quenched or annealed. In the quenched setting, the GMC realization is prescribed, and we show that the phases are governed by the most probable Hölder exponent of the multifractal velocity field. In the annealed setting, the GMC is averaged over, leading to an additional smoothing effect. Moreover, we show that the separation process exhibits structural analogies with multiplicative one-dimensional versions of the Liouville Brownian motion (LBM)--a diffusion process evolving in a random GMC landscape, originally introduced in the context of Liouville quantum gravity. In particular, both the quenched and annealed phase transitions are recovered by considering a multiplicative LBM characterized by a roughness parameter and an intermittency exponent .

Paper Structure

This paper contains 40 sections, 17 theorems, 105 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Phase transitions in the 1D Kraichnan model
  3. The regularized separation process
  4. Characterization as a diffusion process
  5. Characterization as a time-changed Brownian motion
  6. The unregularized separation process
  7. Definition
  8. Boundary behaviors
  9. Phases of the Lagrangian flow
  10. The Bessel shortcut to the phase diagram
  11. The limit of vanishing regularization
  12. Multifractal Kraichnan flows
  13. Gaussian multiplicative chaos
  14. The regularized multifractal separation process
  15. The unregularized multifractal separation process
  16. ...and 25 more sections

Key Result

Lemma 2.1

Let $\xi\in(0,2]$. There exist constants $C_+,C_->0$ such that, for all $r \in[-L,L]$,

Figures (7)

  • Figure 1: Top panels show the scaling coefficient $\mathcal{A}$, with numerical estimates of the prefactors $c_d(r)$ given by Eq. \ref{['eq:cd']} in insets for $\xi =2/3$ (left) and $\xi=4/3$ (right). The bottom panels show the corresponding RV kernels. Numerics use $\eta=10^{-10}$ and $\psi(x) = 1/2-\tanh(4x-1.5)/2$
  • Figure 2: Left: Effective parameters $a$ and $d_e$ of the dual Bessel process \ref{['eq:bessel']}. Right: Regularized norms $|\cdot|_{\eta,\beta}$ for $\eta =0.2$ and various $\beta>0$
  • Figure 3: Left: $\eta$-regularization of the log kernel \ref{['eq:log-kernel']} used in Kahane's martingale approximation. Right: Random realizations of the regularized GMC $\mu_\eta = (\eta/e)^{2\gamma^2} e^{2\gamma \Gamma_\eta}$ for $\eta=10^{-3}$ (deep blue) and $\epsilon = 10^{-6}$ (pale blue). Inset shows the regularized Gaussian field $\Gamma_\eta$ with similar conventions.
  • Figure 4: Left: Compensated moments of the diffusion coefficient ${\mathbb E}^{\Gamma}\left(\mathcal{A}^{(\gamma)}(r)^p\right)^{1/p}r^{-\xi} \propto r^{\zeta_A(p)/p-\xi}$ obtained by Monte Carlo sampling using $2^{22}$ grid points, for $\gamma=0.2$, $\xi =2/3$ and $p=1/64, 1/32, 1/16, 1/8,1/4$ (from top to bottom). Right: Same as the left panel, but for $p=-1/64, -1/32, -1/16, -1/8, -1/4$ (from top to bottom).
  • Figure 5: Phases of the Lagrangian flow in the multifractal Kraichnan model. Left: Quenched setting of Theorem \ref{['thm:multifractal phases']}. Right: Annealed setting of § \ref{['ssec:annealed']}.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • Definition 3.1: Multifractal Kraichnan separation process
  • ...and 20 more