On the Relationship between Radiative Entropy and Temperature Distributions

The Earth can be viewed as a complex nonequilibrium system that exchanges primarily radiative energy and entropy with its surroundings. The energy balance equation provides an important constraint on the distribution of outgoing radiation since the net global energy exchange will be close to zero over some appropriately long time interval. The entropy of the radiation does not obey such a conservation law, instead the outgoing entropy irradiance is much greater than the incoming amount. Most of this increase in the entropy flux is due to the conversion of short wavelength photons from a small solid angle into longer wavelength photons that are emitted nearly isotropically. If the entropy irradiance is calculated with sufficient precision, it is possible to relate it to the distribution of radiative temperature over position, direction, wavenumber and polarization spaces. The radiative entropy decreases as the variance of the radiative temperature distribution increases over any of the four spaces listed above, under the constraint of a constant total irradiance. A uniform temperature distribution produces the maximum entropy flux for a given energy flux. Calculations were performed for simple Earth-like temperature distributions with the result that each of the radiative temperature distributions over position, direction and wavenumber produce a 0.1%–1.5% decrease in the total entropy irradiance compared to the uniform equilibrium case. Meridional temperature gradients, longwave emission/absorption lines and surface to atmosphere temperature differences all result in nearly the same magnitude in the lowering of the outgoing entropy flux compared to the isothermal equilibrium condition. Calculations suggest that the current radiation energy budgets from satellites may be usefully augmented by including the entropy irradiance. Time series of the global outgoing longwave entropy irradiance is shown to be significantly different from the global outgoing longwave radiation, even though the calculations are derived from the same data. These results indicate that the dynamical activity of the atmosphere can be remotely monitored in terms of its distance from equilibrium.