Greenhouse effect

An atmosphere that absorbs and reemits thermal radiation can reduce outgoing energy at a given surface temperature, requiring a warmer surface to balance absorbed stellar energy. The planetary equilibrium temperature refers to the radiation budget, not the resulting surface temperature. Water, carbon dioxide and clouds can alter this balance.

A single-line greenhouse deficit measures the reduction of outgoing thermal radiative flux by one isolated absorption line. Consider a plane-parallel atmosphere in local thermodynamic equilibrium, without scattering, above a blackbody surface. Let pressure $p$ increase from zero at the top to $p_s$ at the surface, and let the surface share the adjacent gas temperature $T_s$.

To specify the angular approximation, use one effective upward direction cosine $0<{\mu }_d\le 1$ and a flux closure $F_{\nu }=\pi I_{\nu }$. Define the corresponding transmission ${\mathfrak{T}}_{\nu }(p)=\exp [-{\tau }_{\nu }(p)/{\mu }_d]$. Integration by parts of the formal radiative transfer solution givesFor a sufficiently narrow absorption line centred at frequency ${\nu }_0$, replace the Planck function and its temperature derivative across the line by their values at ${\nu }_0$. With the line equivalent widththe outgoing radiative flux becomesIf temperature increases downward, $D\ge 0$. An atmospheric thermal inversion can invalidate that sign argument; absorption alone does not guarantee a positive deficit for every imposed thermal profile.

For constant absorber mass fraction $q$, constant gravity $g$ and an opacity ${\kappa }_{\nu }=S{\varphi }_{\nu }$ per absorber mass, assume $\int {\varphi }_{\nu }d\nu =1$ and constant integrated strength $S$. Then ${\tau }_{\nu }=qS{\varphi }_{\nu }p/g$. In the optically thin limit for the specified direction,This leading term is independent of the normalized shape of the absorption spectrum. The angular closure and abundance convention must accompany the coefficient; a full angular integration uses its own flux-weighted transmission.

A black surface receiving absorbed stellar irradiance $Q$, beneath a single non-solar-absorbing cloud with thermal emissivity $e>0$, satisfies $T_c^4=T_g^4/2$ when both bodies are in pure radiative equilibrium. The surface balance is $\sigma T_g^4=Q+e\sigma T_c^4$, giving the displayed result. If instead convection pins $T_c=q{T}_g$, the whole-column balance gives $T_g^4=Q/[\sigma (1-e+e{q}^4)]$; the nonradiative energy budget cannot then be discarded.

A greenhouse gas absorbs and emits planetary thermal radiation and therefore contributes to the greenhouse effect. Its climatic impact depends on absorption spectrum, concentration, lifetime and chemical effects. Carbon dioxide, methane and water vapour are important examples.

The incremental radiative forcing per increment of a specified atmospheric gas burden. State whether the abundance unit is emitted mass, mole fraction or ppb, and specify the radiation-adjustment convention. For equal emitted masses in a fixed atmospheric air inventory, a per-ppb coefficient must be divided by the gas molar mass before comparing efficiencies. Spectral overlap with existing absorption makes efficiency depend on the background state.

Overlap of a gas absorption spectrum with existing atmospheric absorption changes its incremental radiative effect. A strong band in an already opaque spectral interval can contribute less additional outgoing-flux reduction than a weaker band in a window. Pressure broadening, altitude, temperature and the source spectrum also matter.

Global warming potential compares the time-integrated radiative forcing of equal-mass emission pulses of a greenhouse gas and carbon dioxide, over a stated time horizon $H$. It is not an instantaneous temperature ratio, and its numerical value depends on the horizon and included chemical and climatic responses.

For a gas pulse with unchanged greenhouse gas radiative efficiency and decay $e^{-t/\tau }$, its integrated forcing through $H$ is proportional to $\tau (1-e^{-H/\tau })$. Comparing two such lifetimes at fixed carbon-dioxide reference gives the ratio of these two factors. A fractional lifetime increase is not generally the same fractional GWP increase at finite $H$.

A global warming potential compares equal-mass pulses. If ${ϵ}_i$ is the greenhouse gas radiative efficiency per ppb, an emitted mass produces a mixing-ratio increment inversely proportional to molar mass $M_i$. For two gases with exponential pulse removal, their GWP ratio at horizon $H$ is $({ϵ}_i/{ϵ}_j)(M_j/M_i){\tau }_i(1-e^{-H/{\tau }_i})/[{\tau }_j(1-e^{-H/{\tau }_j})]$. The common carbon-dioxide reference integral cancels. Comparing only the per-ppb coefficients would instead compare unequal emitted masses.