Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2019 / 4

The first term describes collision-coalescence: a collector sweeps area $\pi R^2$, overtakes small droplets at relative terminal speed $u_R-u_r$, encounters liquid mass per volume $L$, and retains a fraction $E$ of that intercepted liquid. The second term describes diffusion-limited condensational growth, with vapour supersaturation $S-1$ and combined thermal and vapour-diffusion resistances $F_k+F_d$. Condensation is positive for $S>1$ and reverses to evaporation for $S<1$. The liquid-water content $L$ here is a local mass-per-volume quantity, not a vertically integrated liquid-water path.

For an initially monodisperse cloud the relative settling speed vanishes, so there is no gravitational collection term in this model. Set $D=(S-1)/(F_k+F_d)$ and use $M=4\pi {\rho }_l{r}^3/3$. Condensational growth givesTaking upward height from cloud base as positive, the droplet speed relative to the ground is the updraft minus its downward terminal velocity:For a launch at $h_r(0)=0$,This assumes the prescribed uniform updraft and supersaturation persist along the trajectory, and that the Stokes-type radius-squared fall law remains applicable.

Neglecting $r_0$, the no-settling trajectory is $h_r=Ut$, while the settling trajectory is a downward-curving parabola, $h_r=Ut-k_{r}D{t}^2$. For the given coefficients $k_{r}D=3.0\times 10^{-5}\mathrm{m}{\mathrm{s}}^{-2}$. At $2000\mathrm{s}$,The radius is then $r=\sqrt{2Dt}=31.6\mu \mathrm{m}$, with terminal speed $0.12\mathrm{m}{\mathrm{s}}^{-1}$, still much less than the updraft. Such droplets are carried almost to cloud top rather than falling as rain. Condensation alone would take about $2\times 10^4\mathrm{s}$ to reach $100\mu \mathrm{m}$, much longer than the approximately $2000\mathrm{s}$ cloud transit time. It also reduces relative radius differences because $r^2$ grows at the same rate for every droplet, leaving inefficient differential gravitational collection. This is the warm-rain growth bottleneck, not a claim that condensation is absent.

Neglect condensation and the smaller droplets' fall speed, while retaining their specified liquid-water content. For exponential cloud collector growth, the collection law and spherical collector mass giveThus $R(t)=R_0{e}^{{\alpha }_{R}t}$. The growing terminal speed eventually exceeds the updraft, andwhere the last expression uses $t$ in seconds and $U=1\mathrm{m}{\mathrm{s}}^{-1}$. Numerically,The trajectory reaches maximum height when $k_{R}R=U$, at $t=1145\mathrm{s}$, $R=125\mu \mathrm{m}$ and $h_{\max }=620\mathrm{m}$. It returns to cloud base near $t=1860\mathrm{s}$, with radius about $392\mu \mathrm{m}$. The left-to-right sketch in the preceding figure shows this rise and subsequent fall.

Once the drop leaves the cloud, the prescribed in-cloud $L$, $U$ and collection law need not persist. Consequently the negative $2000$-second value describes the requested constant-coefficient model's extension, not a prediction that collection continues below cloud base. Its actual subsequent height needs an outside-cloud model. Within the cloud, the exponential feedback between increasing radius and increasing swept volume explains how a minority of collector drops can escape the condensation bottleneck and initiate warm rain; depletion, breakup and changes in fall law eventually limit indefinite exponential growth.

A broad initial cloud-droplet size distribution can be seeded by giant or unusually soluble cloud condensation nuclei, producing a minority of larger droplets before gravitational collection becomes efficient. Variable supersaturation histories, entrainment and mixing can broaden the distribution relative to a perfectly homogeneous monodisperse model. Turbulent relative motion and preferential concentration can increase encounters, and rare successful collisions can create unusually rapid stochastic collectors. These mechanisms can work together; a single deterministic condensational law applied equally to identical droplets cannot create the initial differential fall speeds by itself. For warm rain, the subsequent amplification is liquid collision-coalescence, distinct from ice-phase precipitation mechanisms.