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A further refinement would recognize the importance of
convection in the lower atmosphere. The average temperature
profile in the lower atmosphere is exactly that followed by a
parcel of air warmed by contact with the Earth's surface and
then rises adiabatically through the troposphere. That has to
be more than just coincidence.
[snip]
Having said all that, evidence suggests that that
this process [radiative equilibrium] is not dominant in
our atmosphere.
The non-isothermal nearly-isentropic profile of
the troposphere is set by convection, swamping
the effect of radiative transport. (The height
of the tropopause corresponds to the height of
the tallest thunderstorms.) Furthermore, the
stratosphere is pretty much isothermal,
indicating that in the absence of convection,
radiation and re-radiation aren't sufficient to
set up a large temperature gradient.
Are you claiming the magnitude of your effect is
independent of the thickness of the atmosphere?
Unfortunately, CO2 is only a player in a few narrow bands of
the IR radiated upward from the Earth. Within those narrow
bands it is basically opaque and between the bands it is
almost totally transparent. Doubling the CO2 content of the
atmosphere produces no significant change in the IR radiation
budget.