Here's some more charts.
KISS Calculated Incoming Solar Insolation, TOA and surface
KISS Calculated Outgoing Radiation, Surface and TOA
Absorption for extremely short pathlengths
Absorption for moderately short pathlengths
All the diagrams are calculated using the Hitran database and basic thermodynamics.
The absorption for surface emissions indicates that a tremendous portion is absorbed very low in the atmosphere and relatively little is absorbed higher up.
50% of the energy is dissipated by 300m and a whopping 25% is dissipated in less than 40 meters. And, over 4 watts/m^2 is dissipated in the first 10cm above the surface. All this is for the current molecular concentrations.
Over the entire atmosphere, the difference between current molecular concentrations and current with 2x co2 (from now) amounts to 3.2W/m^2. At the far low end, differences rise to as much as 6w/m^2 due to additional absorption of 2xco2 but at higher altitudes, this differences diminishes down to the point that above 1km, there's very little difference in the dissipation per height segment between the current and the co2 doubled value. In other words, the dissipation occurs only in the lower atmosphere and there is actually less dissipation of outgoing energy for the 2x co2 than for current concentrations by a very small amount.
This only includes surface radiation and does not include disspation or heat flow from reradiation or from convection. In any case, there is a very significant heat flow going on already with normal concentrations. This also tends to be only about a 3% increase in the heat absorption in the first 1km.