Point-source scalar turbulence

The statistics of a passive scalar randomly emitted from a point source is investigated analytically for the Kraichnan ensemble. Attention is focused on the two-point equal-time scalar correlation function, a statistical indicator widely used both in experiments and in numerical simulations. The only source of inhomogencity/anisotropy is the injection mechanism, the advecting velocity being here statistically homogeneous and isotropic. The main question we address is on the possible existence of an inertial range of scales and a consequent scaling behaviour. The question arises from the observation that for a point source the injection scale is formally zero and the standard cascade mechanism cannot thus be taken for granted. We find from first principles that an intrinsic integral scale, whose value depends on the distance from the source, emerges as a result of sweeping effects. For separations smaller than this integral scale a standard forward cascade occurs. This is characterized by a Kolmogorov-Obukhov power-law behaviour as in the homogeneous case, except that the dissipation rate is also dependent on the distance from the source. Finally, we also find that the combined effect of a finite inertial-range extent and of inhomogeneities causes the emergence of subleading anisotropic corrections to the leading isotropic term, that are here quantified and discussed.