  The statistics of a passive scalar randomly emitted from a point source is
investigated analytically. Our attention has been focused on the two-point
equal-time scalar correlation function. The latter is indeed easily related to
the spectrum, a statistical indicator widely used both in experiments and in
numerical simulations. The only source of inhomogeneity/anisotropy is in the
injection mechanism, the advecting velocity here being statistically
homogeneous and isotropic. Our main results can be summarized as follows. 1)
For a very large velocity integral scale, a pure scaling behaviour in the
distance between the two points emerges only if their separation is much
smaller than their distance from the point source. 2) The value we have found
for the scaling exponent suggests the existence of a direct cascade, in spite
of the fact that here the forcing integral scale is formally set to zero. 3)
The combined effect of a finite inertial-range extension and of inhomogeneities
causes the emergence of subleading anisotropic corrections to the leading
isotropic term, that we have quantified and discussed.
