An Answer For Lumo and Neutrino

More on lapse rates of planetary atmospheres.

Consider the following one-dimensional thought experiment. Let the surface be at temperature Ts and the elevated radiating level of the atmosphere be a temperature Te < Ts. Suppress convection for the moment (perhaps by replacing it by glass of equivalent optial properties). Consider a layer of the atmosphere somewhere above the surface. It absorbs upwelling radiation from below and downwelling radiation from above, and the amount absorbed is proportional to its absorption coefficient alpha. It also emits radiation half downward and half upward proportional to alpha and the Planck function. Because it is hotter below, it absorbs more from below and less from above. If you do the integration, you get that its radiative equilibrium temperature is given by Ts^4 = (Te^4)*(1+xi0), where xi0 is the integrated opacity from point e to the surface. If you make the crude but reasonable assumption that opacity (from z to ground) is given by xi = xi0*exp[-z/He] you get a profile of the radiative equilibrium temperature for the atmosphere T = Ts/(1+xi0*exp[-z/He])^(1/4).

Put in the numbers for the Earth and you get that the lapse rate in the lower 11 kilometer is greater than the adiabatic lapse rate - so the bottom ll km of the atmosphere is convectively unstable - which drives tropospheric lapse rates toward adiabatic.

Make xi0 much smaller, and the radiative lapse rate would be less than the adiabatic lapse rate, in which case the atmosphere would be stable and subadiabatic.

This simplified model is developed (with all the detailed steps) in Richard P. Wayne's Chemistry of Atmospheres ppg 44-56 in my second edition copy. More accurate results from detailed computer models show the same pattern: Radiative equilibrium sets the table, convection smooths the curve when adiabatic rates are exceeded.

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