Perspectives on scaling and multiscaling in passive scalar turbulence

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.