  We obtain an implicit equation for the correlation dimension which describes
clustering of inertial particles in a complex flow onto a fractal measure. Our
general equation involves a propagator of a nonlinear stochastic process in
which the velocity gradient of the fluid appears as additive noise. When the
long-time limit of the propagator is considered our equation reduces to an
existing large-deviation formalism, from which it is difficult to extract
concrete results. In the short-time limit, however, our equation reduces to a
solvability condition on a partial differential equation. We show how this
approach leads to a perturbative expansion of the correlation dimension, for
which the coefficients can be obtained exactly and in principle to any order.
We derive the perturbation series for the correlation dimension of inertial
particles suspended in three-dimensional spatially smooth random flows with
white-noise time correlations, obtaining the first 33 non-zero coefficients
exactly.
