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A change in cloud microphysics or cloud properties caused by aerosol, including droplet activation, precipitation changes and adjustments of water content or coverage. The Twomey effect isolates a fixed-water-content optical response; it does not by itself establish the sign or magnitude of every cloud adjustment.

Condensed water falling from clouds or deposited from the atmosphere as liquid or solid particles. It can efficiently scavenge soluble gases and aerosol particles. Gas conversion into sulfate and subsequent atmospheric precipitation removal are separate stages of an atmospheric sulfur budget.

A suspension of liquid-water droplets, ice particles or both in an atmosphere. Clouds interact with solar and thermal radiation; their net radiative effect depends on height, water content, particle size distribution and coverage.

The fraction of incident radiation reflected by a cloud under a specified spectral and angular convention. For the approximate shortwave relation $A=\tau /(\tau +7)$, increasing optical depth raises albedo with diminishing sensitivity near $A=1$. Transmitted and absorbed radiation must also be accounted for in the energy budget.

For $\tau \propto N_d^{1/3}$ at fixed liquid-water content and thickness, and $A=\tau /(\tau +7)$, the derivative is $dA/d{N}_d=A(1-A)/(3N_d)$. For a fixed fractional number increase, the initial response is greatest at $A=1/2$. The exact finite-change expression is $A^{\prime }=sA/(1-A+sA)$ with $s=(N_d^{\prime }/N_d{)}^{1/3}$. Uncertain activation of additional particles and changes in water content limit direct application to real clouds.

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.

A liquid-water particle suspended in a cloud. Number, size distribution and liquid-water content are separate properties; increasing number at fixed total water shifts the characteristic radius downward.

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 lost ozone-column photolysis shielding is $\Delta N=20(2.69\times 10^{16})=5.38\times 10^{17}\mathrm{molecules}\mathrm{c}{\mathrm{m}}^{-2}$, so its optical-depth decrement is $\sigma \Delta N=0.0538$. Under the stipulated effective-cross-section Beer-Lambert model, the transmitted photodissociation flux increases by $\exp (0.0538)$.

The short-lived oxygen singlet D atom satisfies $[\mathrm{O}({}^{1}D)]\simeq J_{{\mathrm{O}}_3}[{\mathrm{O}}_3]/(k_q[M])$ when quenching dominates. At fixed local ozone, water and air density, hydroxyl radical production is proportional to $J_{{\mathrm{O}}_3}$. ConsequentlyA linear optical-depth approximation would give $5.38\%$. A realistic calculation integrates the wavelength-dependent absorption cross section, quantum yield and actinic flux; clouds and multiple scattering modify the simple transmitted-beam model.

Ozone change can also alter the radiative temperature structure, water vapor and circulation, thereby changing the water reaction channel and the supply of ozone/precursors to the troposphere. If the held-fixed tropospheric ozone assumption is relaxed, altered ozone itself changes the photochemical source of oxygen singlet D atom. These are additional feedbacks, distinct from the isolated UV-transmission calculation above.

Carbon dioxide is relatively well mixed, its abundance is measured accurately, and its spectroscopy and large-scale radiative effect are well constrained. Aerosols are short lived and spatially heterogeneous; uncertain emissions, removal, size, composition, water uptake and height change both scattering and absorption. Their aerosol-cloud interaction effects and subsequent cloud adjustments add uncertainty, and the unobserved preindustrial aerosol baseline complicates anthropogenic attribution. Thus total aerosol forcing combines poorly constrained physical pathways and a less certain counterfactual, rather than simply an uncertain molecular absorption coefficient.^[ref]

First, weakly absorbing sulfate scatters shortwave electromagnetic radiation and tends to cool the surface/planetary energy budget, with magnitude and sign influenced by cloud background and particle optics. Second, sulfate can supply cloud condensation nuclei: at fixed cloud water, more activated droplets give smaller droplets and greater cloud albedo through the Twomey effect; atmospheric precipitation and water-content adjustments can modify that idealized cooling. Third, sulfate particles change atmospheric composition through uptake, aqueous chemistry and heterogeneous surface reactions, with their acidity and available reactants controlling what is converted. Particle surfaces alone do not guarantee polar-style stratospheric chlorine activation in an ordinary tropospheric cloud.^[ref]

For independent first-order sinks acting on the same well-mixed burden, loss frequencies add:The fraction converted to sulfate before removal by the competing direct-deposition sink isThe individual event fractions are $0.40$ for direct sulfur dioxide deposition, $0.40$ for cloud oxidation and $0.20$ for hydroxyl radical oxidation. These are branching fractions for the sulfur dioxide fate; sulfate formed by oxidation can subsequently be deposited as well. The argument assumes the supplied channel lifetimes are compatible effective first-order rates, not residence times of isolated air masses that never exchange.

For a liquid cloud at fixed water content and geometrical thickness, more cloud condensation nuclei can produce more, smaller droplets. The increased total extinction area raises optical depth and reflected sunlight: the Twomey effect gives negative shortwave radiative forcing. It is an indirect aerosol effect mediated by cloud microphysics, rather than direct aerosol scattering.^[ref]

Smaller droplets can also delay collision-coalescence and drizzle, potentially increasing liquid-water retention, cloud lifetime or coverage; that adjustment would usually strengthen cooling for low reflective clouds. It is not universal: evaporation and mixing with dry air can reduce water content or cloud cover, and dynamical changes can oppose the simple lifetime argument. Absorbing particles can heat an atmospheric layer, sometimes evaporating cloud and producing a positive semi-direct forcing, but their vertical position can instead stabilize a cloud layer and reverse the response. Aerosols can additionally alter ice nucleation and mixed-phase precipitation, with no single forcing sign independent of cloud type. The net response combines shortwave reflection and longwave thermal effects, and depends on cloud height, optical depth and adjustments. Thus the fixed-water Twomey sign is robust within its assumptions; adjustment and ice-cloud signs require context.^[ref]

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.

For a small fractional change $ϵ=\Delta N_d/N_d$, use $\Delta A\simeq A(1-A)ϵ/3$. With $ϵ=0.1$ the changes areThus the respective albedos become approximately 0.5083 and 0.7070 in the first-order calculation. These are increases of 0.83 and 0.70 percentage points, not 0.83% and 0.70% relative changes. Relative to the initial albedos, the increases are about 1.67% and 1.00%.

The exact finite-change version of the stipulated optical model is $A^{\prime }=sA/(1-A+sA)$ with $s=(1.1{)}^{1/3}$, giving $A_{\mathrm{marine}}^{\prime }=0.50794$ and $A_{\mathrm{continental}}^{\prime }=0.70663$. Its slightly smaller changes are consistent with curvature neglected in the differential approximation. For equal fractional number increases, the cleaner, intermediate-albedo marine cloud is somewhat more susceptible; for equal absolute number increases its susceptibility would be much greater because of the extra $1/N_d$ factor. The two starting albedos are treated as the supplied separate baselines, not as an exact common-proportionality calibration of the approximate optical-depth law.

Hold liquid-water content and cloud thickness fixed, as required for a partial sensitivity to droplet number. Then $\tau =K{N}_d^{1/3}$, so $d\tau /d{N}_d=\tau /(3N_d)$. Differentiating the albedo relation gives $dA/d\tau =7/(\tau +7{)}^2$, henceSince $A=\tau /(\tau +7)$ and $1-A=7/(\tau +7)$,The factor $A(1-A)$ peaks at $A=1/2$, so the albedo response to a fixed fractional droplet-number change is greatest at intermediate albedo and weaker near either saturation limit.

Cloud changes caused by aerosol absorption of radiation and the resulting heating. Heating can evaporate clouds and reduce shortwave reflection, but the effect can reverse when heating changes stability and cloud maintenance differently. The forcing sign depends on the relative positions of cloud and absorbing aerosol and on dynamical adjustment.

A wintertime region of strong circumpolar flow that can isolate polar stratospheric air and sustain cold conditions. Isolation, cloud formation, sunlight return and subsequent breakup jointly influence seasonal ozone chemistry. Cooling alone does not determine the full ozone evolution.

Atmospheric particles containing sulfate, commonly in sulfuric-acid/water or neutralized-salt mixtures. Particle size, water uptake, acidity and accompanying material determine scattering, absorption, cloud activation and heterogeneous chemistry; sulfate mass alone does not specify radiative forcing.

Chapter 7 of Climate Change 2021: The Physical Science Basis, pp. 923–1054. DOI 10.1017/9781009157896.009. Cited sections: emissions metrics and aerosol–cloud interactions; this institutional assessment is not an answer set for the Cambridge paper.