= Solution
{c}
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 are
$$
\sigma T_g^4=Q+e\sigma T_c^4,\qquad
e\sigma T_g^4=2e\sigma T_c^4.
$$
For $e>0$ this gives
$$
\boxed{T_g=T_0(1-e/2)^{-1/4},\qquad T_c=T_0(2-e)^{-1/4}.}
$$
The $e\to0^+$ 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 $e=1$:
$$
\boxed{T_g=2^{1/4}T_0\simeq1.189T_0.}
$$
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 \b[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 requires
$$
H=e\sigma(2T_c^4-T_g^4),\qquad
Q+e\sigma T_c^4-\sigma T_g^4=H.
$$
For 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.
Back to article page
