Lagrangian Supersaturation Fluctuations at the Cloud Edge

Evaporation of cloud droplets accelerates when turbulence mixes dry air into the cloud, affecting droplet-size distributions in atmospheric clouds, combustion sprays, and jets of exhaled droplets. The challenge is to model local correlations between droplet numbers, sizes, and supersaturation, which determine supersaturation fluctuations along droplet paths (Lagrangian fluctuations). We derived a statistical model that accounts for these correlations. Its predictions are in quantitative agreement with results of direct numerical simulations, and explain the key mechanisms at play.

Figure 1

(a) Initial condition used in Refs. [16, 17, 18, 19, 29]: a three-dimensional slab of cloudy air with supersaturation $s_{\mathrm{c}}\ge 0$, containing droplets (filled black circles) with initial number density $n_0$, surrounded by dry subsaturated air with $s_{\mathrm{e}}<0$ (hashed). The supersaturation profile is shown as a solid line. The cloudy air occupies a volume fraction $\chi$. (b) Statistical-model rate $\phi (t)$ as a function of time for different Damköhler numbers; see Eq. (4) and SM [30] for details. Parameters: $\chi =0.4$, ${\mathrm{Da}}_{\mathrm{s}}=0.08$, ${\mathrm{Da}}_{\mathrm{d}}=0.0073$; ${\mathrm{Da}}_{\mathrm{s}}=0.80$, ${\mathrm{Da}}_{\mathrm{d}}=0.073$; ${\mathrm{Da}}_{\mathrm{s}}=8.0$, ${\mathrm{Da}}_{\mathrm{d}}=0.73$ (solid lines). Parameters from Ref. [17], but the value of ${\tau }_L$ in our DNS differs slightly from that in Ref. [17] due to statistical variability in the forcing. The dotted line is the steady-state limit ${\phi }_{*}=C_{\varphi }/2$ for a passive scalar with $C_{\varphi }=2$ [34] (see text).

Figure 2

Lagrangian supersaturation distributions. (a) ${\mathrm{Da}}_{\mathrm{s}}=0.80$, ${\mathrm{Da}}_{\mathrm{d}}=0.073$, $\chi =0.4$ (parameters from Ref. [17]). DNS results (see SM [30] for details): symbols ($t=0.68$, blue filled square; $t=1$, green filled triangle; $t=1.69$, red filled circle; $t=2.36$, orange filled diamond). Statistical-model simulations: solid lines. Statistical-model simulations for a passive scalar ($t=0.68$, 2.36, dashed). The shift due to phase change is indicated by horizontal arrows. (b) Same as (a) but for ${\mathrm{Da}}_{\mathrm{s}}=8.0$, ${\mathrm{Da}}_{\mathrm{d}}=0.73$ [17]; passive-scalar result shown only for $t=0.68$. The DNS results were obtained using the same turbulent velocity field and the same initial conditions.

Figure 3

Correlations between droplet-number density and supersaturation. Average $⟨n(t)|s(t)=s⟩$ of droplet-number density at time $t$ conditional on local supersaturation $s$. (a) ${\mathrm{Da}}_{\mathrm{s}}=0.80$, ${\mathrm{Da}}_{\mathrm{d}}=0.073$, and $\chi =0.4$ (parameters from Ref. [17]). DNS results (see SM [30] for details): symbols ($t=0.68$, blue filled square; $t=1.69$, red filled circle; $t=2.36$, orange filled diamond). Statistical-model simulations (lines). (b) Same as (a) but for ${\mathrm{Da}}_{\mathrm{s}}=8.0$ and ${\mathrm{Da}}_{\mathrm{d}}=0.73$ [17].