Transport in Multifractal Kraichnan Flows: From Turbulence to Liouville Quantum Gravity

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 $$\xi \in (0,2]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , controlling the correlation decay for the Gaussian component, and the intermittency parameter $$\gamma \in [0,\sqrt{2}/2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , prescribing the deviations from self-similarity. Recent numerical work by the authors suggests that such coupling induces 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 $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> and $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> , 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—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 $$\xi + 4\gamma ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>γ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> and an intermittency exponent $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> .