= Radiative-convective climate column
A <radiative-convective climate column> applies a <radiative-convective equilibrium model> to the vertical thermal structure relevant to <climate change>. <Radiative transfer> determines heating and cooling; <convective adjustment> represents unresolved vertical redistribution when the chosen <atmospheric lapse rate> is too steep. <Radiative equilibrium> alone need not give a physically admissible lower-atmospheric profile.
In 1967, <Syukuro Manabe> and <Richard T. Wetherald> computed equilibrium with prescribed <relative humidity>, allowing atmospheric <water vapor> to change with <temperature>. Their average-cloud case warmed by $2.36\,\mathrm K$ for <carbon dioxide> increasing from 300 to 600 parts per million, versus $1.33\,\mathrm K$ with fixed absolute humidity. These are model outputs, not measured warming. The comparison isolates an important <water-vapor feedback>.<Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity by Syukuro Manabe and Richard T. Wetherald>[Manabe and Wetherald, 1967, abstract; Section 4b, Table 5 and Fig. 16]{ref}
An original two-temperature calculation exposes the bookkeeping. Let an opaque surface absorb solar <irradiance> $Q$ and emit <blackbody> <thermal radiation> $\sigma T_s^4$. A single isothermal atmospheric layer transmits all sunlight and absorbs a fraction $\epsilon$ of surface infrared emission, emitting $\epsilon\sigma T_a^4$ both upward and downward. This is a grey thermal-band <emissivity>, with $0<\epsilon\leq1$. Let $H$ be upward nonradiative <heat flux>, and let $C_s,C_a>0$ be <heat capacities> per unit <area>. Then <conservation of energy> gives
$$
C_s\dot T_s=Q+\epsilon\sigma T_a^4-\sigma T_s^4-H,
\qquad
C_a\dot T_a=\epsilon\sigma T_s^4-2\epsilon\sigma T_a^4+H.
$$
The two equations sum to
$$
\frac{d}{dt}(C_sT_s+C_aT_a)=Q-L,
\qquad
L=(1-\epsilon)\sigma T_s^4+\epsilon\sigma T_a^4.
$$
Here $L$ is <outgoing longwave radiation>. Both $H$ and downward atmospheric emission cancel internally: they redistribute <energy> without creating it. The factor two in atmospheric emission counts two faces. Dropping the downward face would spoil the surface budget. Each right-hand term has units $\mathrm{W\,m^{-2}}$; each $C$ has units $\mathrm{J\,m^{-2}\,K^{-1}}$.
Choose $Q=240\,\mathrm{W\,m^{-2}}$, $H=80\,\mathrm{W\,m^{-2}}$ and $\epsilon=1$. At <equilibrium>, $L=\sigma T_a^4=240\,\mathrm{W\,m^{-2}}$ and $\sigma T_s^4=400\,\mathrm{W\,m^{-2}}$, giving $T_a\simeq255.06\,\mathrm K$ and $T_s\simeq289.81\,\mathrm K$. Direct substitution gives surface balance $240+240-400-80=0$ and atmospheric balance $400-480+80=0$. These deliberately chosen fluxes illustrate the model; they are not an observed global budget or a reconstruction of the 1967 calculation. Prescribing $H$ illustrates <heat transfer> accounting but does not close a realistic <convection> law.
\Image[/climate-column-energy-and-spectral-readout.png]
{title=A conserving <climate model> column and the route from outgoing <radiance> to a spectral comparison}
{description=The left panel uses the stipulated opaque-layer fluxes in watts per square metre. The right panel is a functional measurement diagram, not an observed spectrum. Broadband energy accounting and a directional spectral readout test different parts of the description.}
For fixed $Q,H,\epsilon$, the positive-<temperature> <equilibrium> is locally stable. Put $a=4\sigma T_s^3>0$ and $b=4\epsilon\sigma T_a^3>0$. The <Jacobian matrix> of the temperature equations has <trace> $-a/C_s-2b/C_a<0$ and <determinant> $ab(2-\epsilon)/(C_sC_a)>0$. Thus both <eigenvalues> have negative real part. Temperature-dependent <climate feedbacks> alter this test. A single grey layer also omits the <optical spectrum>, cloud structure and the independent thermal behavior of the <troposphere> and <stratosphere>.
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