  The paper studies the behavior of the trajectories of fluid particles in a
compressible generalization of the Kraichnan ensemble of turbulent velocities.
We show that, depending on the degree of compressibility, the trajectories
either explosively separate or implosively collapse. The two behaviors are
shown to result in drastically different statistical properties of scalar
quantities passively advected by the flow. At weak compressibility, the
explosive separation of trajectories induces a familiar direct cascade of the
energy of a scalar tracer with a short-distance intermittency and dissipative
anomaly. At strong compressibility, the implosive collapse of trajectories
leads to an inverse cascade of the tracer energy with suppressed intermittency
and with the energy evacuated by large scale friction. A scalar density whose
advection preserves mass exhibits in the two regimes opposite cascades of the
total mass squared. We expect that the explosive separation and collapse of
Lagrangian trajectories occur also in more realistic high Reynolds number
velocity ensembles and that the two phenomena play a crucial role in fully
developed turbulence.
