Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2021 / 2

Write $\kappa =R_d/c_p$, where $R_d$ is the specific gas constant and $c_p$ the specific heat capacity at constant pressure. Conservation of potential temperature on a dry adiabatic process givesLet $Q=(1-{\alpha }_c)S$ be the absorbed stellar irradiance, and let $T_0=(Q/\sigma {)}^{1/4}$, with $\sigma$ the Stefan-Boltzmann constant. The transparent gas contributes no thermal radiation. For a cloud with emissivity $e$, Kirchhoff's law of thermal radiation gives thermal absorptivity $e$; assume thermal transmissivity $1-e$ and negligible thermal reflection, as in the usual one-layer grey body model. The outgoing radiant flux is the transmitted surface emission plus upward cloud emission:ThusNo factor of four is needed: $S$ is the incident irradiance of the specified column, rather than a stellar-beam irradiance that still needs a spherical average.

For fixed ${\alpha }_c,p_c,p_s$, increasing $e$ monotonically warms the ground when $0<q<1$. At $e=0$, $T_g=T_0$; at $e=1$, $T_c=T_0$ and $T_g=T_0/q$. A cloud at the surface, $q=1$, produces no such warming. Comparing with a genuinely cloud-free column also requires including its different albedo: relative to $(S/\sigma {)}^{1/4}$, the ratio is $[(1-{\alpha }_c)/(1-e+e{q}^4){]}^{1/4}$. Reflection can outweigh the greenhouse effect.

The governing balance here is the whole-column radiation budget. Imposing an additional purely radiative surface balance would incorrectly remove the convective heat transfer that the prescribed dry adiabatic process may require.

The cloud absorbs $e\sigma T_g^4$ from below and emits $e\sigma T_c^4$ from each face. It absorbs no stellar radiation. Its net radiative heating per unit area is thereforeFor $e>0$, a cloud with $q^4<1/2$ warms in the absence of other energy transfers; one with $q^4>1/2$ cools. For the thin pressure thickness $\delta p$, its initial temperature tendency is approximately $g{\mathcal{R}}_c/(c_p\delta p)$ if its sensible heat capacity dominates. Latent heat and changing condensate modify that conversion from heating to temperature change, but not the sign of the radiative input.

With $T_g$ fixed, pure radiative equilibrium requiresThis equals the ambient dry-adiabatic process temperature precisely whenFor ordinary dry air, $\kappa \simeq 2/7$, so the ratio is about $0.545$. If $e=0$, the cloud exchanges no thermal radiation and its radiation budget does not select any equilibrium temperature.

At fixed ground temperature, warming the cloud increases its downward radiant flux $e\sigma T_c^4$, giving a positive contribution to the ground's radiative heating. That is the immediate radiative effect. A self-consistent final answer must also remove the convective heat transfer used to pin the original cloud to the dry adiabatic process.

For pure radiative equilibrium of both bodies, with no net nonradiative energy exchange, their two balances areFor $e>0$ this givesThe $e\to 0^{+}$ limit gives the selected limiting cloud temperature; at exactly $e=0$, only $T_g=T_0$ is fixed radiatively. The maximum greenhouse effect at fixed $Q$ occurs at $e=1$:The pinned dry-adiabatic process case instead has maximum factor $1/q=(p_s/p_c{)}^{\kappa }$. It exceeds $2^{1/4}$ for a high, cold cloud with $q<2^{-1/4}$, equals it at the level found in part b, and is smaller for a low, warm cloud.

In particular, for $q^4<1/2$ the final purely radiative ground is cooler than the pinned solution, even though a warmer cloud increases downward radiation at fixed $T_g$. To see why these statements are compatible, let $H$ be the net upward nonradiative energy transfer from the ground to the cloud in the pinned state. Its conservation of energy equation requiresFor the cold cloud, $H<0$: maintaining it on the assumed dry adiabatic process requires removing its net radiative heating and returning energy elsewhere. Setting $H=0$ changes that energy redistribution. It is therefore incorrect to infer the final ground temperature by changing only the downward radiant flux. Indeed the final cloud is warmer than the pinned one for $q^4<1/2$ and $0<e<1$, while the final ground is cooler. At $e=1$ the cloud temperature is $T_0$ in both whole-column equilibria.

Fixed cloud properties are an artificial constraint. Warming a saturated cloud increases the saturation pressure, tending to evaporate its cloud droplets; the released water vapor, changing latent heat budget and altered buoyancy can change its altitude and emissivity. Cooling favours condensation and potentially atmospheric precipitation. A warm perturbation that survives evaporation can rise by convection, but neither a unique vertical displacement nor survival of the cloud follows without moisture supply and an environmental temperature profile. Finally, pure radiative equilibrium of the two bodies does not require the intervening transparent gas to remain on the original dry adiabatic process; imposing that extra constraint is generally inconsistent except at the matching level in part b.