1. Global Change: Class Exercise
Global Energy Balance
&
Planetary Temperature
Mteor/Agron/Envsci/Envst 404/504

2. Zero-Dimensional, Global Energy Balance Model
Zero-dimensional: steady, time-average
averaged over all spatial directions
Global: average is over the entire planet
Energy balance: balance of incoming and outgoing energy flux
Global energy balance: Radiative equilibrium (in = out)

3. Zero-Dimensional GEBM
Incoming (absorbed) radiation:
At photosphere surface, solar flux ~ W-m-2
At Earth’s orbit, solar flux ~ 1360 W-m-2

4. Zero-Dimensional GEBM
Scattering:
air molecules, aerosols
Reflection: clouds
Planetary Albedo
Surface albedo

5. Zero-Dimensional GEBM
Incoming (absorbed) radiation: R
Incoming = (1-albedo) x (area facing sun) x S
= (1-)pR2S
Emitted radiation:
Outgoing = I x (4pR2) R

6. Zero-Dimensional GEBM
Global energy balance: Radiative equilibrium (in = out)

7. Zero-Dimensional GEBM
Global energy balance: Radiative equilibrium (in = out)
Temperature ??

8. Zero-Dimensional GEBM
Suppose black-body emission:
What then is TRAD?

9. Zero-Dimensional GEBM
Observed, average surface temperature = Ts = 288 K
Why is Ts ≠ TRAD?

10. Zero-Dimensional GEBM
Alternative – imperfect emission:
black-body emission × emissivity
What then is the emissivity ε?

11. Zero-Dimensional GEBM
If CO2 doubles instantaneously, outgoing radiation drops 4 W-m-2 for the same Ts. What is the new ε?
What is the new Ts that restores balance?
How much warming occurs?
(Note that this computation does not include any changes in atmospheric water vapor or snow/ice cover under warming.)

12. Zero-D GEBM: Empirical I
The previous computation does not include any changes in atmospheric water vapor or snow/ice cover under warming. An empirical alternative that includes effects of water vapor changes with temperature is …
I = A + BTs(˚C) , where
A = 214 W-m-2 , B = 1.6 W-m-2-˚C-1

13. Zero-D GEBM: Empirical I
If instantaneous CO2 doubling reduces I by 4 W-m-2 , and the change leads to Anew = 210 W-m-2 , then if there is no albedo change (no change in mean snow/ice cover) …
Ts,new = ?
Ts,new – Ts,old = ?

14. Zero-D GEBM: Albedo Feedback
For global temperature changes of no more than a few degrees, observational evidence suggests that the albedo (the portion of sunlight reflected) varies approximately with temperature as  = 0.32 – 1.33x10-3 Ts
What now is the energy balance equation?

15. Zero-D GEBM: Albedo Feedback
What then is Ts=? (what equation?)
We have S/4 = 340 W-m-2, and B = 1.6 W-m-2-˚C-1.
For current CO2 levels, A = 214 W-m-2, and for doubled CO2 levels, A = 210 W-m-2.
Ts,new = ?
Ts,new – Ts,old = ?

16. End - Class Exercise: Global Energy Balance & Planetary Temperature