A Condensation–coalescence Cloud Model for Exoplanetary Atmospheres: Formulation and Test Applications to Terrestrial and Jovian Clouds

Our model provides the vertical distributions of the mass and number densities of condensate particles (Figure 1). We adopt a 1D Eulerian framework in which the gas ascends at a vertical velocity w. Each condensate particle is assumed to fall relative to the upwelling gas at a terminal speed ${v}_{{\rm{t}}}$, which is given by the balance between gas drag and gravity and is therefore a function of the particle radius r (see Equation (23) for the expression of ${v}_{{\rm{t}}}$ adopted in this study). Thus, each particle has a net vertical velocity $w-{v}_{{\rm{t}}}(r)$. We divide the population of condensate particles into small "cloud particles" whose net vertical motion is upward, $w-{v}_{{\rm{t}}}\gt 0$, and large "rain particles" whose net vertical motion is downward, $w-{v}_{{\rm{t}}}\lt 0$. In this paper, we assume that initial cloud particles form through heterogeneous nucleation (see Section 2.3 for details).

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We denote the total number density and mass density of the cloud (rain) particles by ${N}_{{\rm{c}}}$ (${N}_{{\rm{r}}}$) and ${\rho }_{{\rm{c}}}$ (${\rho }_{{\rm{r}}}$), respectively. We assume that the cloud and rain particles have characteristic radii ${r}_{{\rm{c}}}$ and ${r}_{{\rm{r}}}$ and characteristic masses ${m}_{c}=(4\pi /3){\rho }_{\mathrm{int}}{r}_{c}^{3}$ and ${m}_{r}=(4\pi /3){\rho }_{\mathrm{int}}{r}_{r}^{3}$, respectively, where ${\rho }_{\mathrm{int}}$ is the internal density of the particles. For liquid particles, ${\rho }_{\mathrm{int}}$ is equal to the material density of the condensate, ${\rho }_{{\rm{p}}}$, while for solid particles, ${\rho }_{\mathrm{int}}$ can be lower than ${\rho }_{{\rm{p}}}$ because an aggregate of solid particles can be porous. The porosity of condensate particles can potentially affect the growth and motion of the particles, as demonstrated by theoretical studies on dust evolution in protoplanetary disks (e.g., Ormel et al. 2007; Okuzumi et al. 2009). We here neglect this effect by assuming constant bulk density, ${\rho }_{\mathrm{int}}={\rho }_{{\rm{s}}}$, but we plan to take it into account in future work.

Our model determines the vertical distribution of ${N}_{{\rm{c}}}$, ${\rho }_{{\rm{c}}}$, ${N}_{{\rm{r}}}$, ${\rho }_{{\rm{r}}}$, and the condensate vapor mass density ${\rho }_{{\rm{v}}}$ by numerically solving the set of vertically 1D transport equations with terms representing condensation and coalescence (Sections 2.2 and 2.4–2.6). Condensation refers to particle growth through the accretion of supersaturated vapor, while coalescence refers to the growth through collisions with other condensate particles under gravity. In this study, we refer to the coalescence between cloud and rain particles as sweepout, in order to distinguish the coalescence of two cloud particles or of two rain particles.¹ We neglect the condensation of vapor onto rain particles because the timescale of condensation growth is longer than the timescale of sedimentation for these large particles. The characteristic masses and radii of two particle species (cloud and rain) are automatically determined by the mass and number densities via the relations ${m}_{j}={\rho }_{j}/{N}_{j}$ and ${r}_{j}={(3{m}_{j}/4\pi {\rho }_{{\rm{p}}})}^{1/3}$, respectively, where $j={\rm{c}}$ for cloud particles and $j={\rm{r}}$ for rain particles. Such a framework is called a double-moment bulk scheme in meteorology (e.g., Ziegler 1985; Ferrier 1994) and a characteristic size method in the planet formation community (e.g., Birnstiel et al. 2012; Ormel 2014; Sato et al. 2016). This characteristic size method allows us to simulate the growth of particles with much less computational time than required with spectral bin schemes where the full size distribution of particles is evolved (Birnstiel et al. 2012; Sato et al. 2016). Our method is particularly useful for studying cloud formation over a wide parameter space.

Because the terminal velocity of particles generally increases as they grow, there is a height $z={z}_{\mathrm{top}}$ at which the net upward velocity of cloud particles reaches zero, i.e., $w-{[{v}_{{\rm{t}}}({r}_{{\rm{c}}})]}_{z={z}_{\mathrm{top}}}=0$. At this height, which we call the cloud top, we convert the cloud particles into rain particles and allow them to fall and continue growing (see Figure 1). Our implementation of the cloud-to-rain conversion is described in Section 2.7.

The transport equations used in our model are given by

$$\begin{eqnarray}&&\displaystyle \frac{\partial {N}_{{\rm{c}}}}{\partial t}=-\displaystyle \frac{\partial }{\partial z}[(w-{v}_{{\rm{t}}}({r}_{{\rm{c}}})){N}_{{\rm{c}}}]-{\left|\displaystyle \frac{\partial {N}_{{\rm{c}}}}{\partial t}\right|}_{\mathrm{coal}}-{\left|\displaystyle \frac{\partial {N}_{{\rm{c}}}}{\partial t}\right|}_{\mathrm{sweep}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\displaystyle \frac{\partial {\rho }_{{\rm{c}}}}{\partial t} & = & -\,\displaystyle \frac{\partial }{\partial z}[(w-{v}_{{\rm{t}}}({r}_{{\rm{c}}})){\rho }_{{\rm{c}}}]\\ & & +{\left(\displaystyle \frac{\partial {\rho }_{{\rm{c}}}}{\partial t}\right)}_{\mathrm{cond}}-{m}_{{\rm{c}}}{\left|\displaystyle \frac{\partial {N}_{{\rm{c}}}}{\partial t}\right|}_{\mathrm{sweep}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {N}_{{\rm{r}}}}{\partial t}=-\displaystyle \frac{\partial }{\partial z}[(w-{v}_{{\rm{t}}}({r}_{{\rm{r}}})){N}_{{\rm{r}}}]-{\left|\displaystyle \frac{\partial {N}_{{\rm{r}}}}{\partial t}\right|}_{\mathrm{coal}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rho }_{{\rm{r}}}}{\partial t}=-\displaystyle \frac{\partial }{\partial z}[(w-{v}_{{\rm{t}}}({r}_{{\rm{r}}})){\rho }_{{\rm{r}}}]+{m}_{{\rm{c}}}{\left|\displaystyle \frac{\partial {N}_{{\rm{c}}}}{\partial t}\right|}_{\mathrm{sweep}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rho }_{{\rm{v}}}}{\partial t}=-\displaystyle \frac{\partial }{\partial z}(w{\rho }_{{\rm{v}}})-{\left(\displaystyle \frac{\partial {\rho }_{{\rm{c}}}}{\partial t}\right)}_{\mathrm{cond}},\end{eqnarray} \tag{ 5 }$$

where ${(\partial {\rho }_{c}/\partial t)}_{\mathrm{cond}}$ is the rate of increase in ${\rho }_{c}$ due to condensation, $| \partial {N}_{{\rm{c}}}/\partial t{| }_{\mathrm{coal}}$ ($| \partial {N}_{{\rm{r}}}/\partial t{| }_{\mathrm{coal}}$) is the rate of decrease in ${N}_{{\rm{c}}}$ (${N}_{{\rm{r}}}$) due to the coalescence of cloud (rain) particles themselves, and $| \partial {N}_{{\rm{c}}}/\partial t{| }_{\mathrm{sweep}}$ is the rate of decrease in ${N}_{{\rm{c}}}$ due to sweepout. The expressions for ${(\partial {\rho }_{c}/\partial t)}_{\mathrm{cond}}$, $| \partial {N}_{j}/\partial t{| }_{\mathrm{coal}}$, and $| \partial {N}_{{\rm{c}}}/\partial t{| }_{\mathrm{sweep}}$ are given in Sections 2.4–2.6.

We assume that cloud particles form at the cloud base through the condensation of vapor onto small refractory grains that already exist in the atmosphere. This process is known as heterogeneous nucleation, and such refractory grains are termed cloud condensation nuclei (CCNs). On Earth CCNs include sea salt, ash, and dust from the land (e.g., Rogers & Yau 1989; Seinfeld & Pandis 2006). Lacking information about CCNs in other planets, including exoplanets, we parameterize them with their number density ${N}_{\mathrm{CCN}}$ and radius ${r}_{\mathrm{CCN}}$. The CCN number density is a particularly important parameter because it determines the number density and maximum reachable size of cloud particles growing through condensation.

In principle, cloud particles may also form through homogeneous nucleation, where molecules in supersaturated vapor spontaneously collide to form initial nuclei. Although homogeneous nucleation is the simplest nucleation process, this occurs only at a supersaturation ratio much larger than unity because of the free energy barrier arising from the surface tension (e.g., Rogers & Yau 1989; Marley et al. 2013). By contrast, heterogeneous nucleation generally occurs when the supersaturation ratio is slightly above unity because the CCNs lower the free energy barrier (Rogers & Yau 1989). However, if there are only a few CCNs available in the atmosphere, homogeneous nucleation would dominate over heterogeneous nucleation (Woitke & Helling 2004). For simplicity, we ignore homogeneous nucleation in the present study, but we plan to include this effect in our future modeling.

The cloud base is defined by the location above which the saturation vapor pressure ${P}_{{\rm{s}}}$ of condensing gas under consideration exceeds the partial pressure in the gas phase. This location is mainly determined by the vertical temperature profile of the atmosphere and weakly depends on the mixing ratio of the condensing gas under the cloud base.

If the cloud particle size ${r}_{{\rm{c}}}$ is much larger than the mean free path of condensing vapor molecules, the rate of increase in ${\rho }_{{\rm{c}}}$ due to condensation is given by (Rogers & Yau 1989)

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\partial {\rho }_{{\rm{c}}}}{\partial t}\right)}_{\mathrm{cond}}=\displaystyle \frac{4\pi {r}_{{\rm{c}}}{N}_{{\rm{c}}}D({\rho }_{{\rm{v}}}-{\rho }_{{\rm{s}}})}{\left(\tfrac{L}{{R}_{{\rm{v}}}T}-1\right)({LD}{\rho }_{{\rm{s}}}/{KT}+1)},\end{eqnarray} \tag{ 6 }$$

where L is the specific latent heat of vaporization, ${N}_{{\rm{c}}}$ is the number density of the cloud particles, K is the coefficient of thermal conductivity of the atmosphere, and D, ${R}_{{\rm{v}}}$, and ${\rho }_{{\rm{s}}}={\rho }_{{\rm{s}}}(T)$ are the diffusion coefficient, specific gas constant, and saturation density of the vapor, respectively. The saturation vapor density is related to the saturation vapor pressure ${P}_{{\rm{s}}}$ by ${\rho }_{{\rm{s}}}={P}_{{\rm{s}}}/{R}_{{\rm{v}}}T$. Equation (6) neglects the effects of surface tension and nonvolatile solutes on the saturation vapor pressure over the particles' surfaces. This is a good approximation for activated condensate particles (Rogers & Yau 1989).

The assumption that the particle sizes are larger than the atmospheric mean free path is justified for the clouds on Earth and Jupiter that we consider in Section 3. The atmospheric mean free path in the terrestrial water clouds (where $T\approx 300\,{\rm{K}}$ and $P\approx 1\,\mathrm{bars}$) and in Jovian ammonia clouds (where $T\approx 130\,{\rm{K}}$ and $P\approx 0.5\,\mathrm{bars}$) is $\sim 0.1\,\mu {\rm{m}}$, which is significantly smaller than the typical particle radius in the clouds (see Figures 3 and 5 in Section 3), However, this assumption is not necessarily valid for clouds in exoplanets. For example, ZnS and KCl clouds on super-Earth GJ 1214b could form at $P\sim 0.01\,\mathrm{bars}$ (Morley et al. 2013) , where the mean free path is $\sim 100\,\mu {\rm{m}}$. Furthermore, the observation of GJ 1214b suggests the presence of high-altitude clouds at $P\sim {10}^{-5}\,\mathrm{bars}$ (Kreidberg et al. 2014; Morley et al. 2015), where the mean free path is as long as ${l}_{{\rm{g}}}\sim 1\,\mathrm{cm}$. When one applies our cloud model to such high-altitude clouds, one should replace Equation (6) by the expression in the free molecular regime (for details, see Woitke & Helling 2003),

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\partial {\rho }_{{\rm{c}}}}{\partial t}\right)}_{\mathrm{cond}}=4\pi {r}_{{\rm{c}}}^{2}{N}_{{\rm{c}}}{C}_{{\rm{s}}}({\rho }_{{\rm{v}}}-{\rho }_{\mathrm{vs}}),\end{eqnarray} \tag{ 7 }$$

where ${C}_{{\rm{s}}}$ is the mean velocity of gas molecules.

Under the assumption that the cloud and rain particle size distributions are narrow, the rate of decrease in N_j ($j={\rm{c}}$) due to coalescence is approximately given by

$$\begin{eqnarray}{\left|\displaystyle \frac{\partial {N}_{j}}{\partial t}\right|}_{\mathrm{coal}} & \approx & \displaystyle \frac{1}{2}\pi {({r}_{j}+{r}_{j})}^{2}{N}_{j}^{2}{\rm{\Delta }}v({r}_{j})E\\ & = & 2\pi {r}_{j}^{2}{N}_{j}^{2}{\rm{\Delta }}v({r}_{j})E,\end{eqnarray} \tag{ 8 }$$

where ${\rm{\Delta }}v$ is the relative velocity between the particles induced by gravitational settling and E is the collection efficiency, defined by the ratio of the effective collisional cross section to the geometric one (see below). The factor 1/2 prevents double counting of the collisions. We have used that the geometric collisional cross section between two similar-sized particles is approximately given by $\pi {({r}_{j}+{r}_{j})}^{2}$. We express the differential settling velocity as ${\rm{\Delta }}v=\epsilon {v}_{{\rm{t}}}({r}_{j})$, where the factor $\epsilon (\lt 1)$ encapsulates the effect of nonzero particle size dispersion. Sato et al. (2016) and Krijt et al. (2016) show that when the differential drift velocity is proportional to the particle size, bulk schemes with $\epsilon \approx 0.5$ best reproduce the results of spectral bin schemes that take into account the full size distribution. In this study, we adopt $\epsilon =0.5$ for arbitrary values of r_j.

The collection efficiency E accounts for the effect of the gas flow around a large particle moving relative to the background gas: the gas flow sweeps aside particles that are aerodynamically well coupled to the gas (see, e.g., Slinn 1974; Pruppacher & Klett 1997, their Chapter 14). For a collision between two particles of radii r and $r^{\prime}$ ($r\gt r^{\prime}$), the collection efficiency can be expressed in terms of the Stokes number

$$\begin{eqnarray}&&\mathrm{Stk}=\displaystyle \frac{{v}_{{\rm{t}}}(r^{\prime} )| {v}_{{\rm{t}}}(r)-{v}_{{\rm{t}}}(r^{\prime} )| }{{gr}},\end{eqnarray} \tag{ 9 }$$

which is approximately the ratio between the stopping time $=\,{v}_{{\rm{t}}}(r^{\prime} )/g$ and crossing time $\sim r/| {v}_{{\rm{t}}}(r)-{v}_{{\rm{t}}}(r^{\prime} )|$ of the smaller particle. When $\mathrm{Stk}\ll 1$, the smaller particle is strongly coupled to the flow around the large particle. To zeroth order, E behaves as $E\approx 0$ at $\mathrm{Stk}\ll 1$ and as $E\approx 1$ at $\mathrm{Stk}\gg 1$ (Rossow 1978). In this study, we evaluate E using a smoother analytic function (Guillot et al. 2014, their Equation (99)):

$$\begin{eqnarray}&&E=\max [0,1-0.42\,{\mathrm{Stk}}^{-0.75}],\end{eqnarray} \tag{ 10 }$$

which vanishes at $\mathrm{Stk}\lesssim 0.3$ and approaches unity at $\mathrm{Stk}\gg 1$ (see Figure 12 of Guillot et al. 2014). This expression assumes that flow around the particle is laminar. If the gas flow is turbulent, the collection efficiency can be higher than assumed here (Homann et al. 2016). We approximate $\mathrm{Stk}$ in E as $\mathrm{Stk}\approx {v}_{{\rm{t}}}({r}_{j})\epsilon {v}_{j}/({{gr}}_{j})$. Because $\mathrm{Stk}$ generally increases with r_j, coalescence occurs only after the particle size exceeds a threshold above which $\mathrm{Stk}\gt 1$. Therefore, the production of precipitating rain droplets through coalescence requires growth beyond this threshold by condensation (Pruppacher & Klett 1997).

Equation (10) applies when the background gas behaves as a fluid, i.e., the particle radius is much larger than the mean free path of the gas molecules. In the opposite case where the gas behaves as a free molecular flow, one may assume E = 1. The free molecular flow regime is expected to govern the collisional growth of particles at high altitudes, where the gas density is low.

Similar to Equation (8), the rate of decrease in ${N}_{{\rm{c}}}$ due to sweepout is given by

$$\begin{eqnarray}&&{\left|\displaystyle \frac{{{dN}}_{{\rm{c}}}}{{dt}}\right|}_{\mathrm{sweep}}=\pi {({r}_{{\rm{r}}}+{r}_{{\rm{c}}})}^{2}| {v}_{{\rm{t}}}({r}_{{\rm{r}}})-{v}_{{\rm{t}}}({r}_{{\rm{c}}})| {N}_{{\rm{r}}}{N}_{{\rm{c}}}E,\end{eqnarray} \tag{ 11 }$$

where the collection efficiency E is given by Equation (10) with $\mathrm{Stk}={v}_{{\rm{t}}}({r}_{c})| {v}_{{\rm{t}}}({r}_{{\rm{r}}})-{v}_{{\rm{t}}}({r}_{{\rm{c}}})| /({{gr}}_{{\rm{r}}})$ (see Equation (9)).

Cloud-to-rain conversion occurs at the height $z={z}_{\mathrm{top}}$, where the terminal velocity of cloud particles ${v}_{{\rm{t}}}({r}_{{\rm{c}}})$ equals the updraft velocity w. Once the terminal velocity of cloud particles exceeds the updraft velocity, we numerically fix the net vertical velocity, $w-{v}_{{\rm{t}}}({r}_{{\rm{c}}})$, to zero and let the cloud particles evolve into rain particles at the rate given by ${t}_{\mathrm{conv}}^{-1}=\beta ({t}_{\mathrm{cond}}^{-1}+{t}_{\mathrm{coal}}^{-1})$, where ${t}_{\mathrm{cond}}^{-1}\equiv (\partial {\rho }_{{\rm{c}}}/\partial t)/{\rho }_{{\rm{c}}}$ and ${t}_{\mathrm{coal}}^{-1}\equiv | \partial {N}_{{\rm{c}}}/\partial t| /{N}_{{\rm{c}}}$ are the rates of growth due to condensation and coalescence, respectively, and β is a numerical factor (see also below). The equations that describe the cloud-to-rain conversion are then given by

$$\begin{eqnarray}&&{\left|\displaystyle \frac{\partial {N}_{{\rm{c}}}}{\partial t}\right|}_{z={z}_{\mathrm{top}}}=-\displaystyle \frac{{N}_{{\rm{c}}}}{{t}_{\mathrm{conv}}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\left|\displaystyle \frac{\partial {\rho }_{{\rm{c}}}}{\partial t}\right|}_{z={z}_{\mathrm{top}}}=-\displaystyle \frac{{\rho }_{{\rm{c}}}}{{t}_{\mathrm{conv}}},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\left|\displaystyle \frac{\partial {N}_{{\rm{r}}}}{\partial t}\right|}_{z={z}_{\mathrm{top}}}=\displaystyle \frac{{N}_{{\rm{c}}}}{{t}_{\mathrm{conv}}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\left|\displaystyle \frac{\partial {\rho }_{{\rm{r}}}}{\partial t}\right|}_{z={z}_{\mathrm{top}}}=\displaystyle \frac{{\rho }_{{\rm{c}}}}{{t}_{\mathrm{conv}}}.\end{eqnarray} \tag{ 15 }$$

Strictly speaking, our cloud–rain two-population model breaks down near the cloud top, where the true size distribution of condensate particles (which is not resolved in our model) cannot be approximated with two peaks that are well separated from each other. The numerical factor β we introduced above arises from the lack of information about the continuous particle size distribution from cloud to rain particles. We find that $\beta \gtrsim 1$–100 causes an oscillating motion of the cloud top that prevents us from obtaining a steady solution. By contrast, $\beta \lesssim 1$ provides a stable steady solution, and we find that the vertical distribution of cloud and rain well below the cloud top is insensitive to the choice of β as long as $\beta \lesssim 1$ (see Sections 3.1 and 3.2). We will adopt $\beta =0.1$ in the test simulations presented in the following section.

The lower boundary of the computational domain is set to the cloud base, which refers to the height above which the saturation ratio $S\equiv {\rho }_{{\rm{v}}}/{\rho }_{{\rm{s}}}$ exceeds unity. To determine the location of the could base, we introduce the mixing ratio of the cloud-forming vapor below the cloud base as an input parameter. At the cloud base, we set the number density and radius of cloud particles to be equal to those of CCNs, ${N}_{\mathrm{CCN}}$ and ${r}_{\mathrm{CCN}}$, respectively. For rain particles, we simply allow them to leave the computational domain, with the (downward) flux determined just above the cloud base.

The upper boundary is taken to be the cloud top, and the boundary conditions at that location are given by Equations (12)–(15). Neither cloud particles nor rain particles are allowed to go above the cloud top because, by definition, the net vertical velocities of the particles vanish at the cloud top.

We focus on trade-wind cumuli, which are relatively shallow water clouds but are yet deep enough to develop precipitation. VanZanten et al. (2011) conducted comparative studies of terrestrial cloud models using the data of trade-wind cumuli in the northwestern Atlantic Ocean obtained from the Rain in Cumulus over the Ocean (RICO) field campaign (Rauber et al. 2007). Following VanZanten et al. (2011), we examine how accurately our cloud model reproduces the vertical cloud distribution from the RICO campaign.

Following Weidenschilling & Lewis (1973), we construct the vertical temperature profile by using the dry adiabatic lapse rate ($g/{c}_{{\rm{p}}}=9.8\,{\rm{K}}\,{\mathrm{km}}^{-1}$, where ${c}_{{\rm{p}}}$ is the specific heat at constant pressure) below the cloud base and the wet adiabatic lapse rate (e.g., Equation (7) of Atreya & Wong 2005) above the cloud base. The vertical pressure profile is calculated by assuming hydrostatic equilibrium (which is a good approximation as long as the updraft motion is slow). In accordance with VanZanten et al. (2011), we fix the surface temperature and CCN number density to $298\,{\rm{K}}$ and $70\,{\mathrm{cm}}^{-3}$, respectively, and adjust the water vapor mixing ratio (or the specific humidity) below the cloud base, so that the cloud base is located at $500\,{\rm{m}}$ from the ground. The updraft velocity is taken to be either $0.9\,{\rm{m}}\,{{\rm{s}}}^{-1}$ or $2.0\,{\rm{m}}\,{{\rm{s}}}^{-1}$, which represent the typical updraft velocities in the dense and diffuse parts of the clouds, respectively, in the simulations by VanZanten et al. (2011, their Figure 5).

We approximate the vapor pressure for liquid water with the Arrhenius function

$$\begin{eqnarray}&&{p}_{{\rm{s}},{{\rm{H}}}_{2}{\rm{O}}}=611\exp \left[\displaystyle \frac{L}{{R}_{{\rm{v}}}}\left(\displaystyle \frac{1}{273\,{\rm{K}}}-\displaystyle \frac{1}{T}\right)\right]\,\mathrm{Pa},\end{eqnarray} \tag{ 16 }$$

where the temperature is in K and $L=2.5\times {10}^{6}\,{\rm{J}}\,{\mathrm{kg}}^{-1}$ (Rogers & Yau 1989). The thermal conductivity K and dynamic viscosity η of air are set to $2.4\times {10}^{-2}\,{\rm{W}}\,{{\rm{m}}}^{-1}\,{{\rm{K}}}^{-1}$ and $1.7\times {10}^{-5}\,\mathrm{Pa}\,{\rm{s}}$, respectively, and the diffusion coefficient D of water vapor in air is set to $2.2\times {10}^{-5}\,{{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}$. These are the values at $273.0\,{\rm{K}}$ and $100\,\mathrm{kPa}$ (Rogers & Yau 1989). The CCN radius ${r}_{\mathrm{CCN}}$ is assumed to be $0.5\,\mu {\rm{m}};$ the results presented below are insensitive to the choice of ${r}_{\mathrm{CCN}}$ as long as it is taken to be smaller than $1\,\mu {\rm{m}}$.

Figures 2 and 3 present the results of our test calculations. In Figure 2, we show the steady-state vertical profiles of the number and mass densities of the cloud and rain particles, for w = 0.9 and $2.0\,{\rm{m}}\,{{\rm{s}}}^{-1}$. To highlight the effects of coalescence, we also show the results of simulations without the coagulation and sweepout terms (see the dotted lines). The sizes of the cloud and rain particles as a function of the height z from the ground are shown in Figure 3. In these simulations, the size of the cloud particles reaches $\sim 100\,\mu {\rm{m}}$ at $z\approx 1500\,{\rm{m}}$ for $w=0.9\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and at $z\approx 2200\,{\rm{m}}$ for $w=2.0\,{\rm{m}}\,{{\rm{s}}}^{-1}$. At these heights, the net vertical velocity $w-{v}_{{\rm{t}}}({r}_{{\rm{c}}})$ of the cloud particles becomes zero, and we convert them into rain particles that are allowed to continue growing as they fall toward the ground.

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Now we compare our simulation results with the RICO flight observations. The light- and dark-gray shaded areas in Figure 2 indicate the 5%–95% and 25%–75% ranges of the observed values, respectively, taken from Figure 8 of VanZanten et al. (2011). Overall, we find that our condensation–coalescence model reproduces the observations to order of magnitude. The predicted values of the cloud and rain densities fall within the 5%–95% range of the observations, except at high altitudes $z\gtrsim 1700\,{\rm{m}}$, where the result for $w=0.9\,{\rm{m}}\,{{\rm{s}}}^{-1}$ considerably underestimates the cloud number density. By contrast, the model neglecting coalescence is found to overestimate the mass density of cloud particles by up to an order of magnitude. Furthermore, this condensation-only model fails to reproduce precipitation because the maximum particle size reachable with condensation is too small to fall against an updraft of $w\sim 1\,{\rm{m}}\,{{\rm{s}}}^{-1}$ (see the dashed lines in Figure 3). With coalescence, cloud particles do grow large enough to start falling as rain particles, as we already described above.

Although coalescence resolves the order-of-magnitude discrepancy between the model predictions and observations, the cloud and rain mass densities are still systematically higher than the averages of the observed values. This indicates that the updrafts that produce the observed clouds entrain dry ambient air (e.g., Pruppacher & Klett 1997). Entrainment reduces the temperature and humidity of the updraft, both of which act to suppress the condensation growth of cloud particles. The suppressed growth in turn leads to a slower decrease in the cloud number density with height, because coalescence takes place only after the particles grow sufficiently large (see Section 2.5). Therefore, a model including entrainment might better reproduce both the number and mass density of cloud particles. However, the modeling of entrainment within a 1D framework necessarily introduces additional poorly constrained free parameters (see, e.g., Pruppacher & Klett 1997, Chapter 12.7, 12.8), which we avoid in this study.

Figures 2 and 3 show that there is a jump in the cloud particle radius and a peak in the cloud mass density at the cloud top. This is an artifact arising from our treatment of the cloud–rain conversion at this location. As emphasized in Section 2.7, our cloud–rain two-population model does not perfectly treat the cloud top, where the two populations in reality merge into a single distribution of particles. In fact, we find that the jump and peak features in ${r}_{{\rm{c}}}$ and ${\rho }_{{\rm{c}}}$, as well as the height of the cloud top, weakly depend on the value of the numerical factor β introduced in our cloud–rain conversion calculations. This is shown in Figure 4, where we plot the vertical distributions of the sizes and mass densities of cloud and rain particles for $\beta =0.1$, 1.0, and 10.0. However, we confirm that this artifact has little effect on the vertical distributions well below the cloud top.

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We also attempt to reproduce the observations of ammonia clouds on Jupiter. Following Ackerman & Marley (2001), we focus on ammonia ice clouds that cover Jupiter's upper troposphere, where $P\sim 0.7\,\mathrm{bars}$ and $T\sim 130\mbox{--}140\,{\rm{K}}$ (e.g., West et al. 1986). In accordance with the measurements by the Galileo probe, we model the vertical temperature profile as $T=166\,{\rm{K}}+{\rm{\Gamma }}(z-{z}_{0})$, where ${\rm{\Gamma }}=-2\,{\rm{K}}\,{\mathrm{km}}^{-1}$ is the lapse rate and z₀ is the height at which $P=1.0\,\mathrm{bars}$ (Seiff et al. 1998), and set the mixing ratio of ammonia gas under the cloud base to be $6.64\times {10}^{-4}\,\mathrm{kg}\,{\mathrm{kg}}^{-1}$ (Wong et al. 2004). The vertical pressure profile is determined under the assumption of hydrostatic equilibrium. We take the updraft velocity and CCN number density as free parameters ranging from 0.1 to $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and from 10³ to ${10}^{8}\,{{\rm{m}}}^{-3}$, respectively. For the vapor pressure of ammonia ice, we use the expression by Ackerman & Marley (2001; see also Ackerman & Marley 2013),

$$\begin{eqnarray}&&{p}_{{\rm{s}},{\mathrm{NH}}_{3}}=\exp \left(10.53-\displaystyle \frac{2161.0}{T}-\displaystyle \frac{86596.0}{{T}^{2}}\right)\,\mathrm{bars},\end{eqnarray} \tag{ 17 }$$

where the temperature is in K. The thermal conductivity of the atmosphere is taken to be $K=9.0\times {10}^{-2}\,{\rm{W}}\,{{\rm{m}}}^{-1}\,{{\rm{K}}}^{-1}$, which is the value for the Jovian atmosphere at $T=134.3\,{\rm{K}}$ (Hansen 1979). The dynamic viscosity of the atmosphere is given by $\eta =6.7\times {10}^{-6}\,\mathrm{Pa}\,{\rm{s}}$ based on the formula for the mixture gas of ${{\rm{H}}}_{2}$ and $\mathrm{He}$ (Woitke & Helling 2003), together with the assumption $T\approx 130\,{\rm{K}}$. The diffusion coefficient of ammonia gas is given by the formula $D=2\eta /(3\rho f)$, where ρ is the atmospheric density and f = 5 for ammonia vapor (Equation (14) of Rossow 1978). Following Ackerman & Marley (2001), we take the bulk density of ammonia ice ${\rho }_{{\rm{p}}}$ to be $0.84\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. As in the previous subsection, the CCN radius ${r}_{\mathrm{CCN}}$ is assumed to be $0.5\,\mu {\rm{m}}$.

We compare the steady-state vertical profiles of the cloud and rain particles obtained from our model with the retrievals by Carlson et al. (1994) based on the infrared observations of Jovian ammonia clouds in the Equatorial Zone and North Tropical Zone from Voyager instrument IRIS. Carlson et al. (1994) retrieved the effective particle radius ${r}_{\mathrm{eff}}$, geometric thickness ${H}_{{\rm{p}}}$, and optical depth τ at the wavelength of $0.5\,\mu {\rm{m}}$, where the effective radius refers to the area-weighted average of the radius of visible particles (see, e.g., Kokhanovsky 2004). We evaluate the effective radius in our simulated clouds as

$$\begin{eqnarray}&&{r}_{\mathrm{eff}}=\displaystyle \frac{\int ({r}_{{\rm{c}}}^{3}{N}_{{\rm{c}}}+{r}_{{\rm{r}}}^{3}{N}_{{\rm{r}}})\exp (-{\tau }_{z}){dz}}{\int ({r}_{{\rm{c}}}^{2}{N}_{{\rm{c}}}+{r}_{{\rm{r}}}^{2}{N}_{{\rm{r}}})\exp (-{\tau }_{z}){dz}},\end{eqnarray} \tag{ 18 }$$

where ${\tau }_{z}$ is the optical thickness above height z. The factor $\exp (-{\tau }_{z})$ accounts for the fact that one can only observe particles residing at ${\tau }_{z}\lesssim 1$. In the calculation of τ and ${\tau }_{z}$, we apply the geometric optics approximation to the extinction cross section at visible wavelengths. The geometric thickness of the simulated cloud is taken to be the distance between the cloud base and the cloud top in this study.

The results are summarized in Figure 5. There, the solid lines show the values of ${r}_{\mathrm{eff}}$, ${H}_{{\rm{p}}}$, and τ from our condensation–coalescence model for different sets of the updraft velocity w and CCN density ${N}_{\mathrm{CCN}}$. The orange horizontal bars indicate the retrievals by Carlson et al. (1994): ${r}_{\mathrm{eff}}=70\mbox{--}100\,\mu {\rm{m}}$, ${H}_{{\rm{p}}}\leqslant 0.3{H}_{{\rm{g}}}$, and $\tau =1.2\mbox{--}2.0$, where ${H}_{{\rm{g}}}=20\,\mathrm{km}$ is the pressure scale height and the range for ${r}_{\mathrm{eff}}$ is based on the interpretation by Ackerman & Marley (2001). We find that the predictions from the condensation–coalescence model satisfy all these observational constraints when the updraft velocity w and CCN number density ${N}_{\mathrm{CCN}}$ are assumed to be $\approx 2\mbox{--}3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and $\approx {10}^{6}\,{{\rm{m}}}^{-3}$ (see the solid black lines), respectively. If ${N}_{\mathrm{CCN}}\gtrsim {10}^{7}\,{{\rm{m}}}^{-3}$ (see the dashed lines), the predicted optical depth falls within the retrieved range only when $w\approx 0.2\mbox{--}0.5\,{\rm{m}}\,{{\rm{s}}}^{-1};$ however, for this range of w, the predicted effective radius is too small to be consistent with the retrieval. If ${N}_{\mathrm{CCN}}\lesssim {10}^{5}\,{{\rm{m}}}^{-3}$ (see the dotted lines), our prediction reproduces the retrieved optical depth only when $w\approx 3\mbox{--}7\,{\rm{m}}\,{{\rm{s}}}^{-1}$, but then overestimates the effective radius from the retrieval.

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The results presented in Figure 5 are little affected by the choice of the β parameter introduced in our cloud-top treatment, as shown in Figure 6. One can see that the predictions converge for $\beta \gtrsim 1$. For $\beta =0.1$, the predicted effective radius and optical depth at $w\gtrsim 0.5\,{\rm{m}}\,{{\rm{s}}}^{-1}$ are higher than the converged values, but the deviation from the converged value is as small as a factor of less than 2.

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We compare the updraft velocity of $2\mbox{--}3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ with other calculations to validate our best-fit value. First, the mixing length theory formulated by Ackerman & Marley (2001) suggested that the updraft velocity at the ammonia cloud region is approximately $1\mbox{--}3\,{\rm{m}}\,{{\rm{s}}}^{-1}$, which corresponds to an eddy diffusion coefficient ${K}_{{\rm{z}}}=(2\mbox{--}6)\times {10}^{8}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$ and a mixing length equal to ${H}_{{\rm{g}}}=20\,\mathrm{km}$. Furthermore, the 2D simulations of the Jovian moist convection show that the updraft velocity at the ammonia cloud region is $0.5\mbox{--}3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ (Sugiyama et al. 2014), shown in the vertical orange bar of Figure 5. Therefore, our best-fit value of updraft velocity of $2\mbox{--}3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ is a realistic value for the Jovian ammonia cloud region.

Unfortunately, there is no direct observational constraint on the CCN number density for Jovian ammonia clouds. However, an important hint for the CCN density can be obtained from the in situ observation of the Jovian atmosphere in a relatively particle-free hot spot by the Galileo probe. The observation showed that a concentration of small particles of a mean radius of $0.5\mbox{--}5\,\mu {\rm{m}}$ and a number density of $1.9\times {10}^{5}\mbox{--}7.5\,\times {10}^{6}\,{{\rm{m}}}^{-3}$ was present at a height where ammonia clouds form (Ragent et al. 1998). It is unlikely that the observed particles had experienced coalescence, because the particles are too small to collide with each other (i.e., for particles smaller than a micron in radius, $\mathrm{Stk}\lt 0.5$ and hence $E\lt 0.3$ in the ammonia-cloud-forming region). Therefore, we can infer that the number density of CCNs, which is approximately equal to the number density of small particles before coalescence sets in, was $\sim {10}^{5}\mbox{--}{10}^{7}\,{{\rm{m}}}^{-3}$ in the hot spot that the Galileo probe entered. Interestingly, this value is consistent with our prediction for the CCN density in the Equatorial Zone and North Tropical Zone (see above). We do not think that this comparison justifies our cloud model, because the CCN density in relatively cloud-free hot spots is not necessarily equal to that in the cloudy Equatorial Zone and North Tropical Zone. Rather, this comparison suggests that the CCN concentrations in the two regions could be similar.