Inhomogeneities and caustics in the sedimentation of noninertial particles in incompressible flows

In an incompressible flow, fluid density remains invariant along fluid element trajectories.
This implies that the spatial distribution of perfect passive tracers in incompressible
flows cannot develop density inhomogeneities if they are not already introduced
in the initial condition. Thus, typically, tracer inhomogeneities in such flows
are explained by particle interactions, or by non-ideality arising e.g. from inertial effects.
However, in certain practical situations, density is measured or accumulated on
(hyper-) surfaces of dimensionality lower than the full dimensionality of the flow on
which motion occurs. A practical situation of this type arises when observing particle
distributions sedimented on the floor of the ocean. In such cases, even if ideal tracers
are distributed uniformly over a finite support in an incompressible flow, advection
in the flow will give rise to inhomogeneities in the observed density. In this paper
we analytically derive, in the framework of an initially homogeneous tracer sheet
sedimenting towards a bottom surface, the relationship between the geometry of the
flow and the emerging distribution. From a physical point of view, we identify the
two processes that generate inhomogeneities to be the stretching within the sheet,
and the projection of the deformed sheet onto the target surface. We point out that
an extreme form of inhomogeneity, caustics, can develop for sheets. We exemplify
our geometrical results with simulations of tracer advection in a simple kinematic
flow, study the generic dependence on the various parameters involved, and illustrate
that the basic mechanisms work similarly if the initial (homogeneous) distribution
occupies a more general region of finite extension rather than a sheet.