Title: Turbulent Mass Inhomogeneities induced by a point-source,
Abstract: We describe how turbulence distributes tracers away from a localized source
of injection, and analyse how the spatial inhomogeneities of the concentration
field depend on the amount of randomness in the injection mechanism. For that
purpose, we contrast the mass correlations induced by purely random injections
with those induced by continuous injections in the environment. Using the
Kraichnan model of turbulent advection, whereby the underlying velocity field
is assumed to be shortly correlated in time, we explicitly identify scaling
regions for the statistics of the mass contained within a shell of radius $r$
and located at a distance $\rho$ away from the source. The two key parameters
are found to be (i) the ratio $s^2$ between the absolute and the relative
timescales of dispersion and (ii) the ratio $\Lambda$ between the size of the
cloud and its distance away from the source. When the injection is random, only
the former is relevant, as previously shown by Celani, Martins-Afonso $\&$
Mazzino, $J. Fluid. Mech$, 2007 in the case of an incompressible fluid. It is
argued that the space partition in terms of $s^2$ and $\Lambda$ is a robust
feature of the injection mechanism itself, which should remain relevant beyond
the Kraichnan model. This is for instance the case in a generalised version of
the model, where the absolute dispersion is prescribed to be ballistic rather
than diffusive.