Analytical radiative fluxes

Water vapor has two roles in Earth’s atmosphere. It is condensable and is a greenhouse gas. The condensable nature implies that the optical depth of water vapor (roughly proportional to the vertically integrated path) is largely governed by temperature following the Clausius-Clapeyron scaling \begin{equation} \tau \sim \exp \left ({-\frac{L_v}{R_vT}}\right). \end{equation} As a greenhouse gas, water vapor absorbs and emits at infrared radiation wavenumbers ($\tilde\nu$) governed by the Planck function $B_{\tilde\nu}$, which, when the Wien approximation is taken, yields \begin{equation} B_{\tilde\nu}(T) \sim \exp\left(-\frac{h c\, \tilde\nu}{k_B T}\right) \end{equation} The two are isomorphic — the same Boltzmann factor $\exp(-E/k_B T)$, differing only in the energy $E$ — because photon occupation and the vapor-to-liquid molecule ratio obey the same statistics. Eliminating temperature between them leaves a power law, \begin{equation} B_{\tilde\nu}\big(T(\tau)\big) \approx B_{\tilde\nu}(T_s)\left(\frac{\tau}{\tau_s}\right)^{\gamma}, \qquad \gamma = \frac{h\nu}{l_v} = \frac{\text{photon energy}}{\text{latent heat per H}_2\text{O molecule}}, \end{equation} where $T_s$ and $\tau_s$ are the surface temperature and column optical depth. The exponent is small — below about 0.4 across the thermal infrared — because the latent heat of water vapor is larger than the energy an infrared photon carries.

What is this power law good for? Longwave transfer obeys Schwarzschild’s equation, \begin{equation} \mathrm{d}I_{\tilde\nu} = \big(\pi B_{\tilde\nu}(T) - I_{\tilde\nu}\big)\,\mathrm{d}\tau, \end{equation} which becomes analytically solvable once $B_{\tilde\nu}$ is written as an explicit function of $\tau$. The power law does this, giving upward and downward fluxes, and from there, one can derive theories for atmospheric cooling, outgoing longwave radiation, downward longwave radiation, etc.

For example the column-integrated radiative cooling at each wavenumber follows as \begin{equation} Q_{\tilde\nu} \approx \pi B_{\tilde\nu}(T_s)\, \Gamma(1+\gamma,\ \tau_s)\, \tau_s^{-\gamma}, \end{equation} with $\Gamma$ the lower incomplete gamma function. In the optically thick limit ($\tau_s \gg 1$) this reduces to a pure power law, $Q_{\tilde\nu} \approx \pi B_{\tilde\nu}(T_s)\,\tau_s^{-\gamma}$, whose derivative with respect to $T_s$ vanishes identically, because $\gamma$ is precisely the ratio of the fractional temperature sensitivities of $B_{\tilde\nu}$ and $\tau_s$. This is an analytical statement of Simpson’s law: at fixed relative humidity, the outgoing longwave radiation at water-vapor wavenumbers does not change as the surface warms.

Because $\gamma$ is built from the latent heat and radiative properties of the condensing gas, the same power law should carry over to other condensable greenhouse gases and other planetary atmospheres.

Water vapor is a major greenhouse gas in the Earth’s atmosphere and its concentration is largely controlled by temperature according to the Clausius-Clapeyron relation. Building on previous theoretical work, we show that, in an idealized radiative-convective equilibrium where temperature decreases with height following adiabatic lapse rates and relative humidity is constant in the vertical, the Planck function and optical depth of water vapor are approximately related by a power law. The exponent γof this power law is given by the ratio of the energy of a terrestrial infrared photon (hν) to the latent heat release associated with the condensation of a single water–vapor molecule (l_v), i.e., γ= hν/l_v. This power law originates from the Boltzmann factor appearing in both the Clausius-Clapeyron relation and Planck’s function. The power law enables analytic solutions of the Schwarzschild equation for upward and downward longwave radiation fluxes, from which atmospheric cooling and outgoing longwave radiation (OLR) follow directly as functions of surface Planck emission and column-integrated optical depth. The nondimensional parameter γencapsulates the dual role of water vapor as both a condensable and a greenhouse gas, and the same power law potentially applies to radiative-convective equilibrium in other planetary atmospheres.

Analytical radiative fluxes

References

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