Fractal Iso-Contours of Passive Scalar in Two-Dimensional Smooth Random Flows

Abstract

A passive scalar field was studied under the action of pumping, diffusion and advection by a 2D smooth flow with Lagrangian chaos. We present theoretical arguments showing that the scalar statistics are not conformally invariant and formulate a new effective semi-analytic algorithm to model scalar turbulence. We then carry out massive numerics of scalar turbulence, focusing on nodal lines. The distribution of contours over sizes and perimeters is shown to depend neither on the flow realization nor on the resolution (diffusion) scale r[subscript d] for scales exceeding r[subscript d]. The scalar isolines are found to be fractal/smooth at scales larger/smaller than the pumping scale. We characterize the statistics of isoline bending by the driving function of the Löwner map. That function is found to behave like diffusion with diffusivity independent of the resolution yet, most surprisingly, dependent on the velocity realization and time (beyond the time on which the statistics of the scalar is stabilized).