= Solution
\b[Optical response at fixed <cloud> structure.] Put $b=1-g$ and write the two-stream <cloud albedo> as $A=b\tau/(2+b\tau)$. At fixed <scattering asymmetry parameter>,
$$
\frac{dA}{d\tau}=\frac{2b}{(2+b\tau)^2},\qquad A(1-A)=\frac{2b\tau}{(2+b\tau)^2},\qquad
\boxed{\frac{dA}{d\tau}=\frac{A(1-A)}\tau}.
$$
At zero <optical depth> the last quotient is understood by its finite limiting value. <Optical depth> counts extinction along a ray. For an approximately homogeneous, monodisperse <cloud>,
$$
\tau=\int_0^hN\sigma_{\rm ext}\,dz=NQ_{\rm ext}\pi r^2h\simeq2\pi Nr^2h.
$$
This uses spherical droplets, independent extinction, representative constant $N,r$, and the <geometric optics> limit in which the droplet radius greatly exceeds the relevant visible wavelength and $Q_{\rm ext}\simeq2$. <Diffraction> supplies part of the extinction cross-section; it is not legitimate to obtain the factor two from geometric interception alone. A size distribution requires its second moment rather than an arbitrary mean radius.
With liquid-water density $\rho_w$ and <cloud> <liquid-water content> $L$, the volume constraint gives $r^3=3L/(4\pi\rho_wN)$. Consequently
$$
\tau=2\pi h\left(\frac{3L}{4\pi\rho_w}\right)^{2/3}N^{1/3},\qquad
\frac{d\ln\tau}{d\ln N}=\frac13+\frac23\frac{d\ln L}{d\ln N}+\frac{d\ln h}{d\ln N}.
$$
This proves
$$
\boxed{\frac{d\tau}{dN}=\frac\tau{3N}\left(1+2\frac{d\ln L}{d\ln N}+3\frac{d\ln h}{d\ln N}\right)}.
$$
Keeping $L,h,\rho_w,g$ fixed isolates the <Twomey effect>:
$$
\boxed{\frac{dA}{dN}=\frac{A(1-A)}{3N}}.
$$
It is a partial response, not a claim that <cloud> water and <cloud> geometry stay unchanged under every real <aerosol> perturbation.
\b[Absolute and fractional susceptibility.] Choose $N_*$ so that $A=1/2$ there, and put $n=N/N_*$. Then
$$
A=\frac{n^{1/3}}{1+n^{1/3}},\qquad
N_*\frac{dA}{dN}=\frac{n^{-2/3}}{3(1+n^{1/3})^2}.
$$
The <cloud albedo susceptibility> per added droplet decreases monotonically. Its small-$n$ asymptote is proportional to $n^{-2/3}$, while its large-$n$ asymptote is proportional to $n^{-4/3}$. Thus a fixed absolute droplet-number addition has much more leverage in a clean <cloud> than in an already droplet-rich <cloud>. The formal divergence as $N\to0$ is not a physical infinite response: fixed water content would require indefinitely large drops, eventually invalidating the optical and <cloud> assumptions.
For a fixed \i[fractional] number addition the response is instead $dA/d\ln N=A(1-A)/3$, maximal at $A=1/2$ and small at either very low or nearly saturated albedo. This distinction is essential when comparing interventions specified by an absolute particle addition with those specified by a percentage increase.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2016-cloud-albedo-susceptibility.png]
{title=Absolute and fractional cloud-albedo responses under fixed liquid-water and thickness assumptions}
\b[Brightening and net radiation.] <Marine cloud brightening> would increase the population of suitable <cloud condensation nuclei>, for example by producing sea-salt <aerosol> from seawater and allowing transport into susceptible marine boundary-layer <clouds>. Activation depends on particle size, hygroscopicity and the <cloud>'s attained supersaturation; adding particles does not guarantee a one-for-one increase of droplets. At fixed <cloud> water, more activated droplets mean smaller radii, larger total extinction area, larger shortwave <optical depth> and more reflected sunlight.
Promising physical conditions combine relatively clean, persistent low <clouds>, appreciable incident sunlight and a dark underlying ocean. The extensive marine stratocumulus regions over the eastern subtropical oceans are examples of this combination. An already polluted, optically saturated <cloud> has little remaining shortwave leverage. Bright <clouds> over snow or ice also add relatively little contrast with the surface. Low <clouds> have temperatures closer to the surface than high <clouds>, so their extra longwave greenhouse effect is generally less able to offset the shortwave cooling. Sunlit seasons and latitudes matter; no shortwave benefit occurs at night.
Schematically, the fixed-cloud shortwave perturbation is $\delta F_{\rm SW}\sim-\mathcal S f_c\delta A$, where $\mathcal S$ is the local incident solar flux and $f_c$ the affected <cloud> fraction. Changes in <cloud> fraction, precipitation, <entrainment>, liquid water and longwave emission must also be evaluated before equating the positive derivative above with a guaranteed net cooling. The derivative identifies susceptibility, not a complete climate-response calculation.
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