Simulating passive scalar advection in rough Kraichnan flows

Abstract

We present a systematic Eulerian study of passive scalar advection in rough two-dimensional Kraichnan flows, covering the full range of velocity roughness exponent $h\in (0,1)$. The advection--diffusion equation is integrated directly using a pseudo-spectral method at resolutions up to $2048^2$ grid points, with a frozen-noise Runge--Kutta scheme consistent with the white-in-time construction of the carrier flow. The simulations recover the duality between advected scalar and advecting flow---the smoother the carrier, the rougher the scalar---together with the predicted scaling laws for the second-order statistics. Once the scaling range is properly identified, the fourth-order flatness anomaly is measured across the whole range of $h$, in agreement with the Lagrangian estimates of Frisch \textit{et al.} (1999) and with the perturbative predictions of Bernard \textit{et al.} (1998) and Pumir \textit{et al.} (1997). Benefiting from the fact that our Eulerian simulations give direct access to the full scalar field, we also examine the probability density functions of scalar increments and the corresponding higher-order statistics, which show systematic departures from Gaussianity and from log-normality, with the strongest deviations manifesting for intermediate values of $h$. A central outcome of this work is a systematic account of how the simulation parameters, in particular the molecular diffusivity, must be adjusted with $h$, providing practical guidelines for reliable simulations; we further show that the residual deviations from the theoretical scaling laws are quantitatively accounted for by the finite spectral representation of the carrier flow. Our analysis also serve as a numerical baseline for simulating more realistic extensions of the Kraichnan model, where the carrier flow is coupled with a Gaussian multiplicative chaos and theoretical developments are limited.