Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 2

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.

Using the adiabatic cloud droplet moments, define $N=\int n(r)dr$ and the mean volume radius by $r_v^3=N^{-1}\int r^{3}n(r)dr$. Summing the liquid mass of spherical droplets givesTherefore, with the shallow-cloud liquid-water content $w=c_{w}h$,The growth is $h^{1/3}$ at fixed number concentration, and the radius scales as $N^{-1/3}$ at fixed water content. The dimensions are consistent: $c_w$ has units $\mathrm{kg}{\mathrm{m}}^{-4}$, so $c_{w}h/({\rho }_{w}N)$ has units ${\mathrm{m}}^3$. The formula is a third-moment radius; a monodisperse distribution is not required. Fixed $N$ per volume is the shallow-cloud approximation, not exact conservation of number per parcel volume during an arbitrarily deep expansion.

Let $r_s^2=N^{-1}\int r^{2}n(r)dr$. The effective droplet radius and shape factor give $r_e=r_v^3/r_s^2$ and $k^{1/3}=r_v/r_e$, so $r_s^2=k^{1/3}{r}_v^2$. If the extinction efficiency is approximately independent of droplet size over the distribution,For visible wavelengths and droplets large compared with the wavelength, geometrical optics makes $Q_{\mathrm{ext}}\simeq 2$ a useful approximation. Retaining it as the specified constant, and taking $N,k,c_w$ constant over the cloud,The $3/5$ comes from integrating $h^{2/3}$; replacing the whole cloud by its cloud-top extinction would incorrectly omit it. The effective-radius relation treats the distribution shape through $k$, without assuming all droplets have the same size. Liquid phase, negligible vertical variation in that shape factor, and negligible departures from the adiabatic water profile are essential assumptions.

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.