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A shallow liquid cloud with $w=c_{w}h$ has liquid-water path $W=c_w{H}^2/2$. If $N$, extinction efficiency $Q$ and shape factor $k$ are constant with height, integrating droplet extinction gives ${\tau }_c=(3/5)\pi Q{k}^{1/3}[3c_w/(4\pi {\rho }_w){]}^{2/3}{N}^{1/3}{H}^{5/3}$. Eliminating $H$ yields $N=[250\sqrt{2}/(243\pi Q^{3}k)]{\rho }_w^2{c}_w^{1/2}{\tau }_c^3/W^{5/2}$. Nonadiabatic water profiles and biased water paths strongly affect the retrieval.

For spherical liquid droplets in the geometric-optics regime, the extinction efficiency is approximately2. Comparing projected extinction area with liquid-water mass gives the displayed relation. At fixed liquid-water path, smaller effective radius means larger optical depth, the basis of the Twomey effect. Absorption spectra, phase and cloud adjustments can invalidate a simple inference from this optical-depth change to net climate forcing.

The local mass of suspended liquid water per unit volume, in $\mathrm{kg}{\mathrm{m}}^{-3}$. Integrating this quantity vertically gives liquid-water path; it is not the total moist-air density.

Clouds reflect shortwave sunlight, usually cooling the planet, and absorb and emit thermal infrared radiation, usually warming the surface relative to clear sky. Their net effect depends on altitude, temperature, optical depth, phase and the underlying surface: thick low clouds often have strong shortwave cooling, while cold high clouds can have a comparatively strong longwave warming effect. Surface and top-of-atmosphere budgets must be distinguished; absorbing radiation within a cloud can heat the atmospheric layer while shading the ground.

Condensation and freezing release latent heat; evaporation and melting absorb it. These processes change buoyancy, stability and convective transport. Entrainment of dry air can evaporate droplets and cool the cloud edge, while precipitation removes condensate and water.

Microphysics controls the radiative properties. For liquid droplets with a fixed liquid-water path W, the cloud optical depth at fixed liquid-water path is approximatelyThus more numerous smaller droplets at fixed W give a brighter, optically thicker cloud: the Twomey effect. Ice habit, phase and particle-size distribution also alter scattering, absorption and precipitation. Cloud-water and lifetime adjustments need not preserve W, so a microphysical perturbation has no universal net radiative sign. ^[ref]

In the adiabatic cloud droplet number retrieval, the independently retrieved liquid-water path fixes the geometrical thickness in the adiabatic model:Define $C_c=(3/5)\pi Q_{\mathrm{ext}}{k}^{1/3}[3c_w/(4\pi {\rho }_w){]}^{2/3}$. Since ${\tau }_c=C_c{N}^{1/3}{H}^{5/3}$, eliminating $H$ givesUse SI units to obtain $N$ in ${\mathrm{m}}^{-3}$, then divide by $10^6$ for $\mathrm{c}{\mathrm{m}}^{-3}$. The sensitivity is substantial: $d\log N=3d\log {\tau }_c-(5/2)d\log W+(1/2)d\log c_w-d\log k-3d\log Q_{\mathrm{ext}}$. Thus a numerical retrieval additionally needs the adiabatic condensation coefficient, shape factor and extinction efficiency. A nonadiabatic water profile, mixed phase, horizontal inhomogeneity or a biased water-path retrieval can produce a large number-concentration bias.

Denote aerosol optical depth by ${\tau }_a$ to distinguish it from cloud optical depth. Retrieve $N$ as in d, collocate it with ${\tau }_a$, and estimate an aerosol-to-droplet susceptibilityA regression should control for water content, updraught and cloud regime; aerosol optical depth is not itself a direct measurement of activated nuclei. For an approximately constant $s_a$ over a factor two in aerosol, $N_2/N_1=2^{s_a}$.

Integrate the supplied cloud albedo susceptibility rather than treating a finite doubling as infinitesimal:For a small response, $\Delta A\simeq A_1(1-A_1)s_a\log 2/3$. If $F_{\mathrm{sw}}$ is the appropriate mean incoming shortwave flux and $f_c$ the fraction of the scene occupied by these clouds, the leading top-of-atmosphere perturbation, positive downward, isThe two transmissions account approximately for the atmosphere above the cloud and may be absorbed into an effective illuminating flux. In the simplest global transparent-overburden estimate $F_{\mathrm{sw}}=L_{⊛}/4$. Positive $s_a$ gives increased reflection and negative shortwave forcing, the Twomey effect. Numerical forcing cannot be determined from the given susceptibility alone: it needs baseline albedo, cloud fraction, illumination and an empirically constrained $s_a$. Humidity swelling, cloud contamination of aerosol retrievals and common meteorological influences can create spurious correlations. Changes in cloud amount, water path or lifetime are additional adjustments, not the fixed-water Twomey term derived here.

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.