Eqn. (3.7) is recovered substituting $\rho$ with $\frac{\rho\hat\rho}{\rho+\hat\rho}$.

**IV. Solid tracers: 1-particle statistics**

We consider the simplest case of a linear drag. In the presence of gravity (or of a constant external force) and of the turbulent velocity field $\u(\x,t)$, the solid particle coordinate $\z^\smalP(t|\x,0)$ will obey the equation of motion: $$\dot\z^\smalP(t|\x,t)=\v^\smalP(\x,t)+\u_G,\qquad \z^\smalP(0|\x,0)=\x
\eqno(4.1)$$ where $\u_G$ is the gravitational drift, that we suppose constant and uniform and $\v^\smalP$ is the fluctuation in the Lagrangian solid particle velocity, which obeys the linear relaxation equation: $$\dot\v^\smalP(\x,t)=\tau_S^{-1}(\u(\z^\smalP(t|\x,0),t)-\v^\smalP(\x,t))
=\tau_S^{-1}(\u^\smalP(\x,t)-\v^\smalP(\x,t)),
\eqno(4.2)$$ with $\tau_S$ the Stokes time. (For a spherical particle of radius $a$ and density $\rho_\smalP$, in a fluid of density $\rho_{\scriptscriptstyle{0}}$ and kinematic viscosity $\nu$, we would have: $\frac{2a^2}{9\nu}|1-\rho_\smalP/\rho_{\scriptscriptstyle{0}}|$; we are disregarding any effect from finite particle Reynolds number [@maxey83]). From now on we shall identify Lagrangian quantities calculated on solid particle trajectories by the superscript $P$.

In general the non-coincidence of fluid and solid particle trajectories makes the analysis of Eqns. (4.1-2) a very difficult task. The short correlation time limit $\rho\to\infty$, however, allows to proceed perturbatively in the fluctuating part of the trajectory separation $\u_Gt+\z(t|\x,0)-\z^\smalP(t|\x,0)$. The physical motivation for this is that, from Eqn. (4.2), $\u_Gt+\z(t|\x,0)-\z^\smalP(t|\x,0)$ fluctuates on timescale $\tau_S$ with velocity 