One Layer Model (e.g. our atmosphere)

We assume the following:

• The temperature of the surface of the planet is constant
• The atmosphere is a shell of negligible thickness
• The atmosphere is isothermal
• All the shortwave radiation that is not reflected reaches the planet's surface
• All the longwave radiation emitted by the planet is absorbed by the atmosphere

Energy Balance in the One Zone Atmosphere

F_o = incident flux
T_s = transmission percentage of short wavelength incoming radiation
T_t = transmission percentage of outgoing long wavelength radiation
F_g = Flux from ground
F_a = Flux from the atmosphere.

How this all works, physically:

1. Incident flux from the sun on the top of our atmosphere F_o is filtered through the atmosphere by its short wavelength transmittence 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) This is the key physics and occurs because we can model the atmosphere as a thin slab
   
   
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.
   
   That is, we previously calculated the equilibrium temperature to be 254K.
   
   With the atmosphere we would now have
   
   

   T_g = 255*(1.6)^1/4 = 288K
   
   
   Note that transmission coefficients are a bit complicated in radiative transfer. They are really calculated in units of Specific Intensity . They are not exactly the same as the energy balance units shown below.
   
   Components of T_s
   
   
   
   Components of T_t