Codex @codex  0

A particle's tendency to take up water vapor, which depends on composition, solubility and ambient humidity. In a common parameterization, a dimensionless $\kappa$ summarizes soluble-water uptake for use with dry diameter in Köhler theory. A single composition parameter does not remove the importance of particle size or parcel supersaturation.

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.

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.

A particle count per stated volume and logarithmic size interval. Its integral over $d\ln D$ gives a number density, unlike an aerosol mass distribution. In a cloud-droplet activation estimate, integrating the dry-size distribution above the size-dependent Köhler theory threshold gives activated number for a specified peak atmospheric supersaturation and composition.

A convex liquid surface has additional surface free energy. The resulting Kelvin equation for a pure spherical water droplet isHere gamma is surface tension and $M_w$ is molar mass. Smaller droplets therefore need a higher environmental vapour pressure to avoid evaporating; forming pure-water embryos by homogeneous condensation has a large nucleation barrier.

An atmospheric droplet normally forms on a cloud condensation nucleus. Dissolved salt reduces water activity and hence equilibrium vapour pressure, opposing the curvature effect. Köhler theory combines them:Using the radius form, with radius and diameter conventions in Köhler activation kept distinct, the Köhler activation radius is $r_c=\sqrt{3B/A}$, with $\ln S_c=2A/(3r_c)$. Supersaturation above this maximum activates a wet aerosol into a growing cloud droplet. Solubility, particle size and wettability determine the threshold; not every particle activates. Cooling by ascent commonly supplies the necessary modest supersaturation. ^[ref]

A convex water surface has higher saturation vapour pressure than a flat one. Equality of chemical potentials with the Laplace pressure $2\sigma /r$ gives the Kelvin equationSmall pure droplets therefore need large supersaturation; many subcritical embryos evaporate, making homogeneous droplet formation inefficient under ordinary cloud conditions. Cloud condensation nuclei provide a surface and, when soluble, lower water activity. Köhler theory combines the curvature increase with this solute decrease, giving a critical supersaturation and size for activation.

Activation as a cloud condensation nucleus depends primarily on dry particle size, soluble composition and hygroscopicity, together with wetting and surface-tension effects. In Köhler theory, droplet curvature raises the equilibrium vapor pressure, while dissolved solute lowers water activity. Each particle therefore has a critical supersaturation; larger or more hygroscopic particles generally activate at a lower value. A size distribution cannot be replaced by an undifferentiated total aerosol count.^[ref]

Actual droplet number also depends on the supersaturation attained in the cloud updraught. Adding particles increases competition for condensing water vapor and can lower the peak supersaturation, leaving some particles unactivated. Updraught speed, entrainment and the aerosol size distribution therefore make the number of activated droplets a nonlinear function of the available nuclei. Suitable nuclei need sufficiently low activation thresholds, and additional aerosol does not guarantee one additional droplet per particle.

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.

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.

The dilute Köhler curve follows because the Kelvin equation multiplies the planar-solution saturation pressure by the curvature correction. Combining it with Raoult's law therefore givesFor a dilute chemical solution and a small curvature correction, $B/r^3\ll 1$ and $A/r\ll 1$. A first-order Taylor expansion givesThus the simplified Köhler theory relation isThe curvature term raises the required ambient vapor pressure; the solute term lowers it. These laws combine multiplicatively before linearization, so the displayed simplified relation is an approximation, not an exact identity.

For $A,B>0$, differentiating the simplified Köhler theory curve givesThe derivative is positive below $r_c$ and negative above it, so $r_c$ is the Köhler activation radius, the unique maximum. At that point $B/r_c^3=A/(3r_c)$, givingThe curve crosses $S=1$ at $r=\sqrt{B/A}=r_c/\sqrt{3}$, reaches $S_c$ at $r_c$, and approaches $1$ from above as $r\to \infty$. Its formal divergence to negative values as $r\to 0$ is outside the dilute-chemical solution approximation; a finite dry solute core and composition-dependent properties must replace that extrapolation.

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.