Irradiated planetary atmosphere

An irradiated planetary atmosphere receives stellar energy from above while internal heat emerges from below. Their competition creates radiative, convective, and sometimes thermally inverted regions.

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 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.

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.

A single-layer greenhouse model represents an atmosphere by one uniform layer that absorbs and emits a fraction of thermal radiation. With infrared emissivity $ϵ$, the layer emits $ϵ\sigma T_a^4$ both upward and downward.

For a planet of Bond albedo $A_B$ orbiting a blackbody star of radius $R_{\ast }$ and temperature $T_{\ast }$, complete global redistribution and negligible internal heat giveUniform dayside-only reradiation replaces $4a^2$ by $2a^2$. The planetary equilibrium temperature describes absorbed stellar power; including an intrinsic cooling flux gives a total effective temperature $T_{\mathrm{eff}}^4=T_{\mathrm{int}}^4+T_{\mathrm{eq}}^4$, distinct from $T_{\mathrm{eq}}$. Neither quantity is necessarily the greenhouse-warmed surface temperature.

A semi-grey atmosphere uses separate mean visible and infrared opacities. Their ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}$ controls where stellar energy is deposited.

The irradiation temperature is defined from the incident stellar flux, commonly by $T_{\mathrm{irr}}=T_{\ast }\sqrt{R_{\ast }/a}$ before redistribution and albedo factors are applied.

The internal effective temperature parametrizes the intrinsic cooling luminosity of a planet through $F_{\mathrm{int}}=\sigma T_{\mathrm{int}}^4$.

An atmospheric thermal inversion is a region where temperature increases outward. In a strongly irradiated atmosphere it can arise when shortwave opacity exceeds thermal infrared opacity and stellar energy is absorbed high in the atmosphere.

An upper atmospheric region above the tropopause in which shortwave heating can produce an atmospheric thermal inversion. Ozone on Earth, and methane and aerosols on giant planets, are examples of absorbers that help heat it. Its positive altitude temperature gradient makes it stable against homogeneous-gas thermal convection.

For the usual semi-grey irradiated atmosphere with downward thermal optical depth $\tau$, an inversion occurs where $f_r{T}_{\mathrm{irr}}^4({\gamma }^2-1)e^{-\sqrt{3}\gamma \tau }>T_{\mathrm{int}}^4$, with $\gamma$ the visible-to-thermal opacity ratio. At negligible intrinsic flux the threshold reduces to $\gamma >1$.

A strong-opacity line forms higher than the nearby continuum. Under the Eddington-Barbier approximation, an outward-decreasing temperature therefore makes an absorption line, while an atmospheric thermal inversion makes the line-forming layer hotter and reverses it into emission.

The radiative-convective boundary is where radiative transport first requires a temperature gradient steeper than the adiabatic gradient, causing convection below.

This model solves a vertical atmospheric pressure-temperature profile consistent with heating, radiative transfer and a convective adjustment where the radiative gradient exceeds the adiabatic temperature gradient. Irradiation, intrinsic flux, gravity and opacity determine the radiative-convective boundary. A one-dimensional model describes a column or average rather than resolving all atmospheric winds.

A constant radiative-diffusion coefficient and constant gravity give $T^4=T_0^4+4AP/g$. Thus ${\nabla }_{\mathrm{rad}}=AP/(g{T}^4)<1/4$ for $T_0>0$. A diatomic ideal gas has adiabatic temperature gradient $2/7$, so this idealized profile never becomes convective. A physical radiative-convective boundary requires additional opacity, flux, or thermodynamic variation.

The radiative temperature gradient is the logarithmic pressure gradient ${\nabla }_{\mathrm{rad}}=d\log T/d\log P$ required to carry the flux by radiation. Convection begins when it exceeds the relevant adiabatic gradient.

The adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=(\partial \log T/\partial \log P{)}_s$. For a perfect gas with constant adiabatic index, ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$.

For a homogeneous dry ideal gas in a hydrostatic adiabat, $dT/dz=-g/C_p$, where $z$ increases upwards and $C_p$ is the specific heat capacity at constant pressure per unit mass. It follows from $dT/dP=1/(\rho C_p)$ and hydrostatic equilibrium. The altitude version of the Schwarzschild criterion is $dT/dz<-g/C_p$ for instability; equality is marginal. Moisture, composition gradients and nonideal equations of state modify the criterion.

A dry parcel in rapid pressure balance with an ambient ideal gas has $d{T}_g/dP=T_g\mathcal{R}/(P{C}_p)$. Ambient hydrostatic equilibrium gives $dP/dz=-Pg/(\mathcal{R}{T}_{\mathrm{env}})$, hence $d{T}_g/dz=-(g/C_p)T_g/T_{\mathrm{env}}$. At a temperature-matched launch point this equals the dry adiabatic lapse rate, supplying the local linear stability test. A substantially hotter parcel does not exactly follow the ambient density's hydrostatic adiabat.