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We consider a passive scalar field under the action of pumping, diffusion andadvection by a smooth flow with a Lagrangian chaos. We present theoreticalarguments showing that scalar statistics is not conformal invariant andformulate new effective semi-analytic algorithm to model the scalar turbulence.We then carry massive numerics of passive scalar turbulence with the focus onthe statistics of nodal lines. The distribution of contours over sizes andperimeters is shown to depend neither on the flow realization nor on theresolution (diffusion) scale $r_d$ for scales exceeding $r_d$. The scalarisolines are found fractal/smooth at the scales larger/smaller than the pumpingscale $L$. We characterize the statistics of bending of a long isoline by thedriving function of the L\"owner map, show that it behaves like diffusion withthe diffusivity independent of resolution yet, most surprisingly, dependent onthe velocity realization and the time of scalar evolution.

Fractal iso-contours of passive scalar in smooth random flows