Atmospheric Physics course of the University of Cambridge (Minor topic, 2013–2014)

Historical Part III Atmospheric Physics papers. Availability of complete worked student solutions has not been established; advertised answer years are skipped when identified. Only solutions are reproduced here, with the original public paper linked from each paper.

Treat the quoted solar spectrum as irradiance normal to the beam, approximately constant over the narrow band. The incident band irradiance is $F_{\perp }=19\mathrm{W}{\mathrm{m}}^{-2}$ and the horizontal value is $F_0=F_{\perp }\sin 30^{\circ }=9.5\mathrm{W}{\mathrm{m}}^{-2}$. The cloud's vertical optical depth is $25(0.4)=10$; including the clear atmosphere gives ${\tau }_v=10.07$. The slant optical depth is ${\tau }_s={\tau }_v/\sin 30^{\circ }=20.14$. Thus the Beer-Lambert law givesThe removed fraction is $1-t_{\mathrm{dir}}=0.9999999982$, almost the full incident horizontal $9.5\mathrm{W}{\mathrm{m}}^{-2}$. Beam-normal surface irradiance is twice the horizontal value. These are direct-beam quantities, not the total diffuse-plus-direct surface flux. If the supplied irradiance were already horizontal, the projection factor would instead be omitted.

The Beer-Lambert law solves the source-free attenuation term $dI/ds=-{\beta }_{\mathrm{ext}}I$ of the radiative transfer equation. Extinction includes absorption and scattering out of that ray. Missing terms are thermal emission and scattering into the ray from other directions, including repeated scattering and surface reflection.

A cloud can remove nearly all direct sunlight while still supplying substantial diffuse downward radiation. Consequently the total surface solar flux can greatly exceed the direct-beam answer, although reflection sends some energy back to space and absorption can heat matter. At $550\mathrm{nm}$ ordinary atmospheric thermal emission is negligible. No unique total flux follows from extinction alone; the single-scattering albedo, angular redistribution and boundary conditions are needed.^[ref]

The cloud heating rate in this band is zero: conservative scattering changes photon directions but transfers no net energy to the cloud matter. Let $F_c=9.5e^{-0.02/0.5}=9.1275\mathrm{W}{\mathrm{m}}^{-2}$ enter its top and $t_c=e^{-20}$. In the stipulated single-scattering escape approximation, the downward and upward outgoing fractions areTheir fluxes are approximately $7.9866$ and $1.1409\mathrm{W}{\mathrm{m}}^{-2}$. With upward-positive net flux, both cloud boundaries have $F_{\mathrm{net}}=-D{F}_c$; henceThe layer heat capacity is $\rho c_v\Delta z=3.30\times 10^5\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. Absorption outside the cloud heats those other layers. Dropping the upward flux would spuriously turn reflection into cloud heating. Neglecting multiple scattering is quantitatively poor at optical depth ten, but conservative multiple scattering still cannot create material heating in this band.

Thick low clouds usually reflect much sunlight and emit infrared at temperatures close to the surface, so their daytime shortwave cooling is strong and their longwave greenhouse contrast relatively small. Cold high clouds can strongly reduce outgoing longwave radiation; thin high clouds reflect less sunlight. Thickness and particle properties matter as well as height.

At night there is no solar term. Warm low-cloud tops commonly cool radiatively, promoting condensation and cloud-driven turbulence that can sustain the layer. A cold, optically thin high cloud can instead absorb more infrared from below than it emits, warming and sublimating and thus shortening its lifetime. A gray-layer diagnostic is absorption minus emission, $ϵ(F_{\uparrow }+F_{\downarrow }-2\sigma T_c^4)$. This is a tendency, not a universal lifetime ranking: thick cirrus can cool at its top, and ascent, entrainment, humidity and precipitation may dominate.

A convex water surface has higher saturation vapour pressure than a flat one. Equality of chemical potentials with the Laplace pressure $2\sigma /r$ gives the Kelvin equationSmall pure droplets therefore need large supersaturation; many subcritical embryos evaporate, making homogeneous droplet formation inefficient under ordinary cloud conditions. Cloud condensation nuclei provide a surface and, when soluble, lower water activity. Köhler theory combines the curvature increase with this solute decrease, giving a critical supersaturation and size for activation.

Warm-rain pathway: condensation first grows activated droplets, but its relative size growth slows. A sufficiently broad size spectrum creates different fall speeds; large drops collect smaller ones by collision-coalescence. Collection efficiency, turbulence, liquid-water supply and residence time control the bottleneck. Rain reaching the ground also requires survival against subcloud evaporation.

Ice pathway: below freezing, saturation vapour pressure over ice is lower than over supercooled liquid. In mixed-phase air, ice can grow by deposition while droplets evaporate: the Wegener-Bergeron-Findeisen process. Ice also collects droplets by riming and combines by aggregation. Adequate ice nuclei, sustained supersaturation and residence time are critical; particles must become heavy enough to fall, with snow sometimes melting to rain. Condensation or deposition alone is not generally sufficient for rapid precipitation.

Droplets remain supercooled because creating an ice embryo costs interfacial free energy: freezing has a nucleation barrier even below the equilibrium melting point. Small droplets also have a low probability of containing an effective ice-nucleating inclusion.

Three heterogeneous routes are immersion freezing around a particle already inside the droplet, contact freezing following a collision with a suitable mineral particle, and deposition nucleation from ice-supersaturated vapour onto a suitable surface. At sufficiently low temperature, homogeneous ice nucleation occurs without a foreign surface. Its onset depends on volume and observation time rather than being a universal exact temperature. No biological ice-nucleating agents are needed for these mechanisms.

Direct effect: nonabsorbing aerosol scatters sunlight, usually increasing reflected shortwave flux over dark surfaces. Absorbing aerosol also heats its atmospheric layer while reducing surface sunlight; the top-of-atmosphere sign depends on its altitude, optical properties and underlying albedo.

Cloud albedo effect: soluble, hygroscopic aerosol readily takes up water and can activate as cloud condensation nuclei. At fixed liquid-water path, more smaller droplets increase optical depth and cloud reflectivity: the Twomey effect. Suppressed collision-coalescence can delay rain and change cloud amount or lifetime, though dynamical adjustments prevent a universal sign.

Semi-direct effect: absorption can alter stability and evaporate clouds, or change entrainment and cloud cover. Insoluble dust may be a poor soluble-droplet nucleus but still scatters/absorbs radiation or nucleates ice; coatings can alter its activation. Solubility, absorptivity and ice-nucleating ability are distinct properties, not mutually equivalent categories.^[ref]

The isentropic potential vorticity is Ertel potential vorticity expressed on constant potential temperature surfaces:Here ${\zeta }_{\theta }$ is the vertical curl of horizontal velocity differentiated on an isentropic surface, and ${\sigma }_{\theta }$ is mass per unit area per unit potential-temperature interval. For smooth inviscid adiabatic motion, $Dq/Dt=0$. Conserved anomalies are advected and deformed, explaining much balanced large-scale weather variability. Diagnosing velocity from PV additionally needs balance, stratification and boundary information. Strong latent/radiative heating, friction and irreversible mixing generate or modify PV, limiting a purely conservative explanation.^[ref]

In a simple northern-hemisphere layer with a fixed upper boundary, climbing a mountain compresses the column. Potential vorticity conservation, $(\zeta +f)/h=\mathrm{constant}$, then reduces absolute vorticity: at nearly fixed latitude the flow acquires anticyclonic relative vorticity. Descending the lee side stretches the column and restores or increases cyclonic vorticity. For an eastward streamline the idealized turning tendency is clockwise in the compressed region and counterclockwise on re-expansion; the resulting disturbance can launch a stationary downstream Rossby wave pattern.

This column argument is an idealization. Stratified air can move around the barrier, isentropic thickness changes need not equal the mountain height, and wave breaking or friction can change PV. Northern-hemisphere signs reverse if $f$ changes sign.

For an approximately conserved isentropic potential vorticity, ${\zeta }_{\theta }+f=q{\sigma }_{\theta }$: increased isentropic layer mass implies more cyclonic vorticity, and decreased layer mass more anticyclonic vorticity at fixed latitude. Equivalently, in the thin-layer analogy, stretching increases and compression decreases absolute vorticity.

The usual summer land-heating picture has a deeper, less stably stratified lower atmosphere and favours a cyclonic thermal low; winter land cooling gives a shallow stable layer and favours an anticyclonic thermal high. These are conditional lower-tropospheric tendencies, not a season-only conservation law. Heating is diabatic and the rougher land surface introduces friction and convergence, so parcel PV need not remain fixed. The sign at every altitude cannot be deduced from “ocean to land” alone without the heating/stability profile and background wind.

Use a uniform westerly Barotropic Rossby wave model on a local beta plane. Its dispersion is $\omega =Uk-\beta k/(k^2+l^2)$. A zonal wavenumber-two pattern at latitude $\varphi$ has $k_2=2/(a\cos \varphi )$, so stationarity requiresFor the zonally varying, meridionally uniform idealization $l=0$, this is $U=\Omega a{\cos }^3\varphi /2$. Meridional structure, deformation radius, mean-flow shear and full spherical geometry change the condition; the latitude and zonal wavenumber alone do not fix a unique observed jet speed.

For a shorter zonal perturbation $n>2$ with the same $l=0$ assumptions,Its phase travels eastward relative to the stationary long wave, but westward relative to the air at speed $4U/n^2$; its packet propagates downstream faster than the mean flow in this idealization. For example $n=6$ gives $c_n=8U/9$ and $c_g=10U/9$.

Within the meandering flow, poleward parcel motion lowers relative vorticity and equatorward motion raises it through $D\zeta /Dt\simeq -\beta v$. Synoptic anomalies can reinforce or oppose the background ridges/troughs according to phase, exchange momentum with it, and transfer energy downstream. Linear superposition alone does not cause mutual growth; stronger interactions require shear, baroclinicity or nonlinear PV advection. No unique amplification rate follows from the supplied information.

In the rotating terrestrial frame, horizontal pressure gradient force and Coriolis force balance in geostrophic balance. Gravity establishes the approximately vertical hydrostatic equilibrium; it is not an additional horizontal force in this balance. With eastward $x$, northward $y$, upward unit vector $\hat{\mathbf{z}}$ and Coriolis parameter $f=2\Omega \sin \varphi$,The geostrophic wind runs along atmospheric isobars, with lower pressure on its left in the Northern Hemisphere.

Interpreting the stated south-to-north gradient as $p_y>0$, $p_y=200\mathrm{Pa}/(2.00\times 10^5\mathrm{m})=10^{-3}\mathrm{Pa}{\mathrm{m}}^{-1}$, while $p_x=0$. Using the external nominal Earth's rotation rate $\Omega =7.292115\times 10^{-5}{\mathrm{s}}^{-1}$ gives the meridional-pressure-gradient geostrophic windBoth zonal velocities are westward; their signs reverse if the supplied gradient is interpreted as a northward decrease. ^[ref]

The validity criteria for geostrophic balance require horizontal acceleration and friction to be small compared with the balancing forces: Rossby number $U/(|f|L)\ll 1$ and slow evolution compared with $1/|f|$. The approximation fails near the equator, on sufficiently small or rapidly changing scales, in strongly curved flow needing appreciable centripetal acceleration, and in the frictional surface layer. Neglecting friction near the ground is the stated idealization.

For an illustrative geostrophic wind with linear drag, use acceleration $-r\mathbf{u}$ with $r>0$. The steady horizontal momentum equation givesDrag reduces the wind and hence its Coriolis acceleration; the wind crosses isobars toward lower pressure. Here it points southwest. The pressure-gradient acceleration points south, Coriolis northwest and drag northeast; the three forces sum to zero. The cross-isobar angle satisfies $\tan \alpha =r/f$ for $f>0$. A turbulent boundary layer has a vertically varying stress rather than necessarily this constant linear drag.

Use spectral radiant exitance per unit wavelength. The Planck law isA hotter blackbody radiates more at every wavelength, its wavelength peak moves to shorter wavelengths according to Wien's displacement law, and its integrated flux rises as $\sigma T^4$ by the Stefan–Boltzmann law. The wavelength convention matters: a spectrum plotted per unit frequency has a different peak transformation.

For planetary equilibrium temperature, assume an isotropic blackbody star, a circular orbit, negligible internal heating, unit surface thermal emissivity and efficient global heat redistribution so the surface has one temperature. The received stellar irradiance is $F_{\ast }=\sigma T_{\ast }^4(R_{\ast }/a{)}^2$. A planet of radius $R_p$ intercepts $\pi R_p^2{F}_{\ast }$ but radiates over $4\pi R_p^2$, giving the projected-area factor in planetary energy balanceWithout the uniform-temperature assumption, this calculation gives $⟨T^4{⟩}^{1/4}$, not a unique arithmetic mean of surface temperature.

With Bond albedo $A=0.05$, the absorbed fraction is $1-A$. Hencea decrease of approximately $3.87\mathrm{K}$. This albedo weights incident stellar light and does not imply thermal emissivity $0.95$.

Define the planetary effective emissivity by outgoing top-of-atmosphere flux $F_{\mathrm{out}}={ϵ}_{\mathrm{eff}}\sigma T_s^4$. Maintaining the same absorbed stellar power with $T_s=312\mathrm{K}$ givesThe greenhouse effect allows thermal radiation to escape from colder layers, so the outgoing flux is less than a black surface at $T_s$ would emit. Downward atmospheric emission contributes to the surface energy balance; this does not violate the top-of-atmosphere energy balance.^[ref]

A single-layer greenhouse model makes the distinction quantitative. Take a shortwave-transparent, isothermal layer with infrared absorptivity and emissivity ${ϵ}_a$, above a surface black in the thermal infrared. Writing $Q=(1-A)F_{\ast }/4$, its surface and atmospheric balances areFor ${ϵ}_a>0$, $T_a^4=T_s^4/2$. Radiation escaping to space is $(1-{ϵ}_a)\sigma T_s^4+{ϵ}_a\sigma T_a^4$, soThus $0.849$ is the planet's effective emissivity; $0.301$ is the layer's infrared emissivity within this particular model. They are not interchangeable. Real atmospheres require wavelength-dependent absorption, a vertical temperature profile and nonradiative heat transport; the measured temperatures alone do not uniquely retrieve an atmospheric optical profile.

A teleconnection is correlated atmospheric variability between separated regions. A principal dynamical pathway is a Rossby-wave teleconnection: a displaced air column conserves potential vorticity, so changing planetary vorticity must be balanced by changing relative vorticity. The background potential-vorticity gradient supplies the restoring mechanism.

In a uniform zonal current $U$ on a beta plane, a simple Barotropic Rossby wave hasThe intrinsic phase propagates westward for $\beta >0$; advection can leave a nearly stationary pattern in the terrestrial frame. Energy and information travel with the group velocity, not necessarily with individual crests or the same air parcels. A jet stream can guide wave packets and link distant ridges and troughs.

For example, persistent anomalous tropical heating can excite wave trains that alter remote pressure, temperature and storm tracks. Tropical Pacific ocean-temperature anomalies can modify that heating, yielding reproducible extratropical patterns. Their phase and strength depend on the background winds, potential-vorticity gradients, damping, season and eddy feedbacks. A correlation pattern is not by itself proof of a unique causal pathway, and not every teleconnection is a single stationary linear wave.^[ref]

A foehn wind is relatively warm, dry downslope air on a mountain's lee side. In the classical condensation pathway, ascending windward air expands and cools at the dry adiabatic lapse rate until the lifting condensation level. Condensation then releases latent heat, reducing the cooling rate to the moist adiabatic lapse rate. Atmospheric precipitation removes condensate, and the descending parcel warms by compression at approximately the dry rate. Its relative humidity falls.

For equal starting and finishing heights, complete condensate removal and a constant moist lapse rate above condensation height $z_c$, the precipitation-driven foehn warming isHere $H>z_c$ is the parcel's maximum height and ${\Gamma }_d>{\Gamma }_m$. The warming is adiabatic compression during descent, not heating by friction. Without condensation or a difference in source air, a reversible dry parcel returning to its original height would regain its original temperature.

Precipitation is not essential to every foehn event. Isentropic foehn drawdown can bring air of higher potential temperature from aloft down the lee slope; turbulent mixing and extra solar heating beneath clear skies can also contribute. The sketch shows the classical rainout pathway rather than asserting a unique mechanism.^[ref]

For dry atmospheric Ertel potential vorticity, use absolute vorticity ${\mathbf{\omega }}_a=\nabla \times \mathbf{u}+2\mathbf{\Omega }$ and potential temperature $\theta =T(p_0/p{)}^{R_d/c_p}$:The conservation statement is Ertel's theorem for smooth, inviscid, adiabatic flow with conservative body forces and a thermodynamic equation of state $p=p(\rho ,\theta )$. Stretching of ${\mathbf{\omega }}_a/\rho$ cancels deformation of $\nabla \theta$ in their material derivative; the remaining baroclinic triple product vanishes because $\nabla p$ lies in the span of $\nabla \rho ,\nabla \theta$. The atmospheric form includes $1/\rho$; the commonly used Boussinesq approximation version has a different normalization.

In a predominantly vertical stratification, $q_E\simeq (f+\zeta ){\partial }_z\theta /\rho$. Thus rotation and stratification jointly label air masses. Material conservation explains how parcels deform and stretch while preserving a useful scalar, supports Rossby waves, and helps diagnose sharp fronts, jets and stratospheric intrusions.

Under an appropriate balance model, potential-vorticity inversion recovers associated wind and temperature fields from a PV distribution together with stratification and boundary data. PV alone does not specify every component of an arbitrary atmospheric flow. Potential-vorticity generation by heating and friction limits conservative parcel reasoning near strong heating, surface stress or mixing.^[ref]