Caustic Activation of Rain Showers

We show quantitatively how the collision rate of droplets of visible moisture in turbulent air increases very abruptly as the intensity of the turbulence passes a threshold, due to the formation of fold caustics in their velocity field. The formation of caustics is an activated process, in which a measure of the intensity of the turbulence, termed the Stokes number St, is analogous to temperature in a chemical reaction: the rate of collision contains a factor $\exp (-C/\mathrm{St})$. Our results are relevant to the long-standing problem of explaining the rapid onset of rainfall from convecting clouds. Our theory does not involve spatial clustering of particles.

Figure 1

Particle collision rate as a function of Stokes number, for three different values of the Kubo number [defined in Eq. (3)]. In these simulations [described in full below Eq. (7)] we used $n={10}^3$, $a=2.5\times {10}^{-4}$, $\eta =0.1/\pi$, $\tau =0.1$, and varied $u_0$ and $\gamma$: $\mathrm{Ku}=1$ (⋄), $\mathrm{Ku}=0.2$ (○), and $\mathrm{Ku}=0.04$ (□). The theoretical curves are (7), with actions $S$ obtained from Fig. 3. The constant $C_a$ in $R_a=C_{a}n{a}^d{u}_0/\eta$ was fitted (see text).

Figure 2

Schematic showing the formation of a caustic. (a) Initial configuration, showing droplet velocity as a function of position along an arbitrary line. (b) This manifold has developed fold caustics. The velocity field is multivalued between the folds.

Figure 3

Rate of caustic formation as a function of $\mathcal{I}\propto {\mathrm{Ku}}^2\mathrm{St}$, showing evidence for “activated” behavior: $\mathrm{Ku}=1$ (⋄), with action $S=0.70$, $\mathrm{Ku}=0.2$, $S=0.18$ (○), and $\mathrm{Ku}=0.04$, $S=0.15$ (□). Also shown is the limiting behavior as $\mathrm{Ku}\to 0$, with action $S=0.14$ (dashed line).