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In planar turbulence modelled as an isotropic and homogeneous collection of2-D non-interacting compact vortices, the structure functions S_p(r) of astatistically stationary passive scalar field have the following scalingbehaviour in the limit where the P\'eclet number Pe -> \infty S_p(r) ~constant+\ln({\frac{r}{LPe^{-1/3}}}) for LPe^{-1/3} << L, S_p(r) ~({\frac{r}{LPe^{-1/3}}})^{6(1-D)} for LPe^{-1/2} << LPe^{-1/3}, where L is alarge scale and D is the fractal co-dimension of the spiral scalar structuresgenerated by the vortices (1/2 <= D < 2/3). Note that LPe^{-1/2} is the scalarTaylor microscale which stems naturally from our analytical treatment of theadvection-diffusion equation. The essential ingredients of our theory are thelocality of inter-scale transfer and Lundgren's time average assumption. Aphenomenological theory explicitly based only on these two ingredientsreproduces our results and a generalisation of this phenomenology to spatiallysmooth chaotic flows yields (k\ln k)^{-1} generalised power spectra for theadvected scalar fields.Comment: 10 pages, RevTex, 2 figures, Revised Version. Submitted to Physical  Rev

Scalings of scalar structure functions in a velocity field with coherent vortical structures