Codex @codex  0

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,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,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 $Q_{\mathrm{ext}}\simeq 2$. 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 $r^3=3L/(4\pi {\rho }_{w}N)$. ConsequentlyThis provesKeeping $L,h,{\rho }_w,g$ fixed isolates the Twomey effect:It is a partial response, not a claim that cloud water and cloud geometry stay unchanged under every real aerosol perturbation.

Absolute and fractional susceptibility. Choose $N_{\ast }$ so that $A=1/2$ there, and put $n=N/N_{\ast }$. ThenThe 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 $N\to 0$ is not a physical infinite response: fixed water content would require indefinitely large drops, eventually invalidating the optical and cloud assumptions.

For a fixed 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.

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 $\delta F_{\mathrm{SW}}\sim -\mathcal{S}{f}_c\delta A$, 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.

Normalization and angular pattern. A scattering phase function describes the distribution of single-scattered power over outgoing solid angle, for a specified incident direction, wavelength and polarization convention. Use the common normalizationThe scattering asymmetry parameter is the mean cosine of the scattering angle: forward scattering has positive $g$, backward scattering negative $g$, and a forward–backward symmetric pattern has $g=0$.

Let $C=E_0(2\pi /\lambda {)}^4[(n^2-1)/(n^2+2){]}^2{r}^6$. The given small-sphere angular intensity is $I=C(1+{\mu }^2)/2$, where $\mu =\cos \Theta$. Its total scattered power isNormalizing givesDimensionally the supplied $I$ is radiant intensity, power per solid angle. If interpreted as detector irradiance at a distance, the appropriate inverse-square distance factor is also needed; it cancels out of the normalized angular pattern.

The polar plot has equal forward and backward lobes, each with radius $3/2$, and lateral values $3/4$. A small nonabsorbing particle behaves as an induced electric dipole. Radiation transverse to the dipole and the average over two orthogonal incident polarizations produce the $1+{\cos }^2\Theta$ form. Its odd angular moment vanishes:As the size parameter $2\pi r/\lambda$ grows, Rayleigh scattering in the electric-dipole approximation no longer suffices. Higher multipoles and interference produce structure in the angular pattern, and large particles generally develop a strong narrow forward diffraction peak. Their $g$ is normally positive, although detailed values and angular features depend on refractive index, size distribution and shape; they are not fixed by radius alone. Large spheres may show rainbows and backscattering structure, while nonspherical ice introduces further angular features.

Optically thick transmission. In a nonabsorbing two-stream or transport approximation, it is the transport optical depth $(1-g)\tau$ that controls the loss of forward directional memory. Using the same idealized albedo closure as in option 1 gives total forward-hemisphere transmittanceFor the same large but finite extinction optical depth,The respective transport-depth factors are $0.15$, $0.30$ and $1$. The water cloud scatters most strongly forward and returns the least light backward. All transmittances tend to zero in the infinite-depth limit. The strictly unscattered direct-beam transmittance is instead $e^{-\tau }$, identical in the three cases: the ranking concerns total transmitted radiation, including forward-scattered light, not that direct component alone. Full radiative transfer can also depend on higher angular moments beyond $g$.

The scattering asymmetry parameter is the mean scattering-angle cosine, $g=(4\pi {)}^{-1}\int P\cos \Theta d\Omega$ in the $4\pi$ normalization. Forward/backward symmetry gives $g=0$; positive $g$ indicates a net forward preference. In a two-stream transport approximation, the reduced scattering optical depth is $(1-g)\tau$.