Basically our atmosphere can be modeled as a thin slab of material of finite temperature:

1. Incident flux from the sun on the top of our atmosphere F_o is filtered through the atmosphere by its short wavelength transmittance T_s , where 0 < t_s < 1.0

2. The flux that reaches the ground is then F_o * T_s

3. That flux is absorbed by the ground and the ground heats up to some temperature (T_g).

4. The ground then re-radiates that heat as outgoing long wavelength IR radiation (F_g).

5. Some of that flux (F_g) is absorbed by the atmosphere through its long wavelength transmittence T_t , where 0 < t_t < 1.0

6. The combination of absorption of short (s) and absorption of long (t) wavelength radiation causes the atmosphere to heat up to a temperature (T_a) and then radiate that temperature away as flux (F_a); 1/2 of it up (into space) and 1/2 of it down (back to ground)
   
   
7. Since as much flux goes out of the system as comes into the system we can set up the following equilibrium conditions.

   Top of the atmosphere: F_o = F_a + T_t *F_g
   
   At the ground: F_g = F_a + T_s *F_o

8. Let F_a = F_o - T_t *F_g and substitute then; eventually get that:
   F_g = F_o *(1+T_s)/(1+T_t)

For our atmosphere: T_s = 0.9; T_t =0.2 F_g = 1.6*F_o which leads to a 30-35K increase in the nominal surface temperature.

Now virtually all of that temperature increase comes from the presence of water vapor. Water vapor is the natural greenhouse gas on the Earth. The addition of CO₂ leads to the augmented or enhanced greenhouse effect.

The figure below shows the important spectral response of these two gases. The right hand curve labeled 255 K is the emission spectrum of the Earth. The arrow represents the wavelength at which most of the Earth's re-radiated energy is emitted. Notice that right of the arrow, the water vapor spectrum is almost entirely black. This means that all of those wavelengths are absorbed by water vapor and this is why water vapor is the dominant greenhouse gas in our atmosphere.

The bottom panel shows the absorption spectrum of CO₂. It does not have large, blacked out areas meaning that it absorbs only over a small range of wavelengths. Unfortunately, that small range of wavelengths coincides exactly with the wavelengths at which the Earth re-radiates most of its absorbed solar energy. It is this unfortunate physical coincidence that makes CO₂ an effective absorber of infrared radiation and, therefore, a contributor to increased atmospheric heating, which in turn leads to increased surface heating.