Perspectives on scaling and multiscaling in passive scalar turbulence

TL;DR

This paper introduces a two-parameter hydrodynamic model for passive scalar turbulence, analyzing multiscaling phenomena and connecting to established models like Kraichnan's, offering insights into turbulence scaling behaviors.

Contribution

It proposes a novel two-parameter continuum model for passive scalar turbulence and uses renormalization group analysis to explore multiscaling and simple scaling regimes.

Findings

Model captures transition between simple scaling and multiscaling.

Results reduce to Kraichnan model in a specific limit.

Framework aids understanding of turbulence scaling behaviors.

Abstract

We revisit the well-known problem of multiscaling in substances passively advected by homogeneous and isotropic turbulent flows or passive scalar turbulence. To that end we propose a two-parameter continuum hydrodynamic model for an advected substance concentration $\theta$, parametrised jointly by $y$ and $\overline y$, that characterise the spatial scaling behaviour of the variances of the advecting stochastic velocity and the stochastic additive driving force, respectively. We analyse it within a one-loop dynamic renormalisation group method to calculate the multiscaling exponents of the equal-time structure functions of $\theta$. We show how the interplay between the advective velocity and the additive force may lead to simple scaling or multiscaling. In one limit, our results reduce to the well-known results from the Kraichnan model for passive scalar. Our framework of analysis should be of help for analytical approaches for the still intractable problem of fluid turbulence itself.