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An aerosol particle able to activate into a growing cloud droplet at the supersaturation attained by a cloud parcel. Activity depends on particle dry size, hygroscopicity and interfacial properties. The number active at a specified supersaturation differs from total aerosol number.

An aerosol particle becomes a freely growing cloud droplet after crossing its Köhler theory growth barrier under the parcel's atmospheric supersaturation history. Counting particles whose critical supersaturation is below a specified peak gives an equilibrium-threshold estimate; calculating that peak requires cooling, water vapor depletion and condensational-growth kinetics. Neither total aerosol number nor dry size alone determines activation.

Growth of liquid cloud droplets by collisions followed by merging. Differences in droplet fall speeds and collection efficiency affect the process. Increasing droplet number at fixed liquid-water content can reduce typical sizes and delay drizzle, but cloud mixing and water-content adjustments prevent a universal precipitation response.

The maximum equilibrium supersaturation along a solution-droplet growth curve. Above this threshold, an aerosol particle can grow into an activated cloud droplet under the conditions of the parcel. In Köhler theory, larger dry diameter or greater hygroscopicity generally lowers the threshold; aerosol number alone does not specify it.

In the dilute radius form $\ln S_{\mathrm{eq}}=A/r-B/r^3$, the curvature coefficient A has length units and B has volume units. Setting its derivative to zero gives the displayed maximum. Ambient supersaturation above it permits activation into a growing cloud droplet. Below it, a sufficiently small wet equilibrium can be stable, while the larger-radius intersection is unstable. The approximation should not be extrapolated into the dry core.

Below cloud base, an initially unsaturated closed parcel expands as ambient pressure falls. In the absence of heat exchange its temperature falls approximately at the dry adiabatic lapse rate $g/c_{p,d}$. Its vapour mixing ratio is conserved, but its saturation mixing ratio decreases strongly as it cools. Hence relative humidity rises until the lifting condensation level is reached. The density normally decreases through expansion; the environmental profile determines the parcel's buoyancy and thus whether ascent can continue.

Near saturation, slight atmospheric supersaturation permits suitable cloud condensation nuclei to activate. Köhler theory describes the competition between curvature, which raises equilibrium vapour pressure, and dissolved solute, which lowers it. Cooling generates supersaturation, while condensation onto the new droplets consumes it. Their balance normally keeps in-cloud relative humidity close to, but not identically at, saturation. The activated droplet number depends on the aerosol population and the maximum supersaturation produced by the ascent.

Above cloud base, condensation lowers the vapour content and raises the liquid-water content. The latent heat released partly offsets expansion cooling, so the moist adiabat has a smaller lapse-rate magnitude than the dry one. In a dilute approximation,A closed adiabatic parcel conserves total water $q_t=q_v+q_l$ and approximately its moist entropy; retaining condensate makes this a reversible-cloud idealization, rather than a precipitating pseudoadiabat. In a shallow cloud, the saturation-mixing-ratio decrease and slow density variation yield nearly constant condensation per height, so liquid-water content is approximately linear above base. Once activation is complete, assuming nearly fixed droplet number per volume gives increasing droplet size as condensate accumulates. Entrainment, continued activation, collision-coalescence and atmospheric precipitation can all violate that simplified description; they are excluded from the stated retrieval model.

Using the adiabatic cloud droplet moments, define $N=\int n(r)dr$ and the mean volume radius by $r_v^3=N^{-1}\int r^{3}n(r)dr$. Summing the liquid mass of spherical droplets givesTherefore, with the shallow-cloud liquid-water content $w=c_{w}h$,The growth is $h^{1/3}$ at fixed number concentration, and the radius scales as $N^{-1/3}$ at fixed water content. The dimensions are consistent: $c_w$ has units $\mathrm{kg}{\mathrm{m}}^{-4}$, so $c_{w}h/({\rho }_{w}N)$ has units ${\mathrm{m}}^3$. The formula is a third-moment radius; a monodisperse distribution is not required. Fixed $N$ per volume is the shallow-cloud approximation, not exact conservation of number per parcel volume during an arbitrarily deep expansion.

Let $r_s^2=N^{-1}\int r^{2}n(r)dr$. The effective droplet radius and shape factor give $r_e=r_v^3/r_s^2$ and $k^{1/3}=r_v/r_e$, so $r_s^2=k^{1/3}{r}_v^2$. If the extinction efficiency is approximately independent of droplet size over the distribution,For visible wavelengths and droplets large compared with the wavelength, geometrical optics makes $Q_{\mathrm{ext}}\simeq 2$ a useful approximation. Retaining it as the specified constant, and taking $N,k,c_w$ constant over the cloud,The $3/5$ comes from integrating $h^{2/3}$; replacing the whole cloud by its cloud-top extinction would incorrectly omit it. The effective-radius relation treats the distribution shape through $k$, without assuming all droplets have the same size. Liquid phase, negligible vertical variation in that shape factor, and negligible departures from the adiabatic water profile are essential assumptions.

At fixed ground temperature, warming the cloud increases its downward radiant flux $e\sigma T_c^4$, giving a positive contribution to the ground's radiative heating. That is the immediate radiative effect. A self-consistent final answer must also remove the convective heat transfer used to pin the original cloud to the dry adiabatic process.

For pure radiative equilibrium of both bodies, with no net nonradiative energy exchange, their two balances areFor $e>0$ this givesThe $e\to 0^{+}$ limit gives the selected limiting cloud temperature; at exactly $e=0$, only $T_g=T_0$ is fixed radiatively. The maximum greenhouse effect at fixed $Q$ occurs at $e=1$:The pinned dry-adiabatic process case instead has maximum factor $1/q=(p_s/p_c{)}^{\kappa }$. It exceeds $2^{1/4}$ for a high, cold cloud with $q<2^{-1/4}$, equals it at the level found in part b, and is smaller for a low, warm cloud.

In particular, for $q^4<1/2$ the final purely radiative ground is cooler than the pinned solution, even though a warmer cloud increases downward radiation at fixed $T_g$. To see why these statements are compatible, let $H$ be the net upward nonradiative energy transfer from the ground to the cloud in the pinned state. Its conservation of energy equation requiresFor the cold cloud, $H<0$: maintaining it on the assumed dry adiabatic process requires removing its net radiative heating and returning energy elsewhere. Setting $H=0$ changes that energy redistribution. It is therefore incorrect to infer the final ground temperature by changing only the downward radiant flux. Indeed the final cloud is warmer than the pinned one for $q^4<1/2$ and $0<e<1$, while the final ground is cooler. At $e=1$ the cloud temperature is $T_0$ in both whole-column equilibria.

Fixed cloud properties are an artificial constraint. Warming a saturated cloud increases the saturation pressure, tending to evaporate its cloud droplets; the released water vapor, changing latent heat budget and altered buoyancy can change its altitude and emissivity. Cooling favours condensation and potentially atmospheric precipitation. A warm perturbation that survives evaporation can rise by convection, but neither a unique vertical displacement nor survival of the cloud follows without moisture supply and an environmental temperature profile. Finally, pure radiative equilibrium of the two bodies does not require the intervening transparent gas to remain on the original dry adiabatic process; imposing that extra constraint is generally inconsistent except at the matching level in part b.

The dew-point depression is $6\mathrm{K}$, soKeeping $T_{D,s}=276.15\mathrm{K}$, $H\simeq 158.5\mathrm{km}$ and $c_p=1004\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ gives approximately $747\mathrm{m}$, consistent with the rounded estimate.

The real cloud base can differ because entrainment changes the parcel's atmospheric mixing ratio and temperature, surface humidity varies, and the environmental pressure profile need not match the assumed parcel profile. Finite cloud-droplet activation requires some atmospheric supersaturation, so the first visible cloud droplets need not occur exactly at the thermodynamic lifting condensation level. A parcel must also actually be lifted that far; environmental static stability can prevent this. The constant-latent heat and first-order Taylor expansion approximations produce smaller numerical differences.

A rising parcel cools, lowering the saturation pressure. Once its water vapor supply exceeds that value, atmospheric supersaturation develops. Homogeneous nucleation of pure water is strongly inhibited at the small atmospheric supersaturations typical of clouds; existing aerosol particles provide cloud condensation nuclei. Their dry sizes and composition, together with the parcel's ascent rate, determine how many cloud droplets activate.

Köhler theory combines the Kelvin equation, which raises equilibrium vapor pressure over a curved surface, with Raoult's law, which lowers it for a chemical solution. In a dilute diameter convention, with dry diameter $D_d$ and wet diameter $D$,Here $s=e_v/E(T)-1$ is atmospheric supersaturation, ${\sigma }_w$ is surface tension, and ${\kappa }_h$ measures aerosol hygroscopicity, not the dry-adiabatic process exponent. The maximum of the Köhler curve is the critical supersaturation. Larger or more hygroscopic particles generally have lower thresholds. Cloud-droplet activation requires overcoming that maximum; total aerosol number is not the activated number.

Given an observed aerosol number size distribution $n_j(D_d)=d{N}_j/d\ln D_d$ for composition classes $j$, first assign each class a hygroscopicity and compute its critical supersaturation. If the parcel reaches maximum atmospheric supersaturation $s_{\max }$, the equilibrium-threshold estimate isFor each fitted lognormal distribution, this is its upper-tail integral; it can be evaluated analytically with an error function or numerically from measured bins.

The value $s_{\max }$ must be calculated, not guessed from the aerosol distribution alone. Prescribe parcel upward velocity $w$, initial pressure, temperature and humidity, then integrate expansion cooling, latent heat release, conservation of total water, and diffusional cloud droplet growth. Schematically,where $a_s,b_s>0$, $G$ includes water vapor diffusion and thermal conduction, and $q_l$ is liquid mass fraction. Rising motion produces atmospheric supersaturation; growing particles consume it. Their competition selects the peak and which particles cross their activation barriers. Assume a well-mixed closed parcel, liquid water, soluble spherical particles, specified mixing state, and negligible collision-coalescence and fallout during activation. Slow kinetic growth, weakly soluble particles or entrainment require a more detailed treatment than the threshold integral.

Let $1<S<S_c$ be the ambient saturation ratio. There are two equilibrium radii: $r_{-}<r_c<r_{+}$. Condensation occurs if $S>S_{\mathrm{eq}}(r)$; evaporation occurs if $S<S_{\mathrm{eq}}(r)$. Therefore droplets with $r<r_{-}$ grow towards $r_{-}$, while those with $r_{-}<r<r_c$ shrink towards $r_{-}$. The smaller equilibrium is stable.

Above $r_c$, droplets with $r_c<r<r_{+}$ still undergo evaporation and ultimately approach $r_{-}$. Droplets with $r>r_{+}$ grow, and the larger equilibrium $r_{+}$ is unstable. Merely exceeding $r_c$ is therefore not sufficient for continued growth at subcritical ambient atmospheric supersaturation. For example, if $(S-1)/(S_c-1)=0.6$, the two intersections have approximately $r_{-}/r_c=0.676$ and $r_{+}/r_c=2.349$; a droplet of radius $1.5r_c$ undergoes evaporation. If instead $S>S_c$, there is no equilibrium barrier in this approximation and droplets activate into growing cloud droplets.

Köhler theory explains activation of cloud condensation nuclei, rather than rapid production of atmospheric precipitation. Diffusion-controlled droplet growth is limited by diffusion of water vapor and removal of released latent heat; approximately $d(r^2)/dt$ is proportional to the ambient atmospheric supersaturation. It slows in radius and tends to narrow a droplet size distribution. The available water vapor, changing temperature and shared depletion of atmospheric supersaturation also invalidate unlimited growth at prescribed $S$. Formation of rain-sized drops normally requires collision-coalescence, while colder clouds can additionally use cloud ice nucleation versus ice growth.

The ambient water vapor partial pressure divided by the equilibrium saturation pressure over a stated reference surface at the same temperature. Relative to flat pure water, atmospheric supersaturation is $s=S-1$. A curved solute-containing cloud droplet generally has a different equilibrium saturation ratio from unity.

Increasing cloud droplet number at fixed liquid-water content and cloud thickness reduces mean droplet size and increases total extinction area, raising shortwave optical depth and cloud albedo. This idealized response produces negative shortwave radiative forcing. Changes in cloud amount, water content and dynamics are separate adjustments.^[ref]