Radiative heating rates

A radiative heating rate is a gas temperature tendency. Positive values mean warming. Heating depends on the divergence of energy flux, not on angle-integrated radiance alone: the flux integral contains the projected-area factor given by the direction cosine.

Let \(F=F_\uparrow-F_\downarrow\) be the upward net flux, integrated over frequency. In a plane-parallel atmosphere, with altitude \(z\) increasing upwards, density \(\rho\), and mass-specific heat capacity \(c_p\),

\[\frac{dT}{dt} = -\frac{1}{\rho c_p}\frac{dF}{dz}.\]

Under hydrostatic balance, \(dp/dz=-\rho g\), this becomes

\[\frac{dT}{dt} = \frac{g}{c_p}\frac{dF}{dp}.\]

These vertical relations omit horizontal flux divergence and spherical-area divergence. Finite differences on pressure levels approximate the derivative; the pressure stencil and boundary treatment affect that approximation.

Optical-depth derivatives

Optical depth increases downwards, so that \(d\tau_\nu/dz=-k_{\mathrm{ext},\nu}\). For the spectral net flux,

\[D_\nu = \frac{dF_\nu}{d\tau_\nu},\qquad \frac{dT}{dt} = \frac{1}{\rho c_p} \int k_{\mathrm{ext},\nu}D_\nu\,d\nu.\]

Total extinction is required here: the absorption fraction is already represented in the flux derivative. Frequency-dependent extinction must be applied before frequency integration, and it must correspond to the optical-depth coordinate used for the derivative. A local optical-depth derivative and a finite difference of flux on pressure levels need not give identical numerical results.

See DISORT and VDISORT core solvers for the discrete-ordinate flux expressions and Radiative heating rates for units, sampling locations, and recipe examples.