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# Equilibrium Temp W Ghg

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Lesson 4 of Climate Change for Advanced Physics; teacher: Gary Bent

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### Equilibrium Temp W Ghg

1. 1. Equilibrium Temperature with Greenhouse Gases
2. 2. T z Equation for temperature change with altitude in the troposphere Γ is called the lapse rate for the troposphere.
3. 3. o C 6 .7 km Average lapse rate for dry air in the troposphere
4. 4. At this altitude, 95% of IR escapes to space without further absorption.
5. 5. We want to find what the altitude is for 95% escape of IR, And the surface temperature of the Earth with greenhouse gases.
6. 6. We will use a two system model of the atmosphere. The two systems being the troposphere and the stratosphere. We will assume thermal equilibrium in the stratosphere.
7. 7. Stratosphere Troposphere
8. 8. 4 I abs Tt
9. 9. 4 I rad (1 ) Tt 4 I abs Tt
10. 10. 4 I rad (1 ) Tt 4 I abs Tt 4 I rad Ts 4 I rad Ts
11. 11. For thermal equilibrium, power in = power out Pin=Pout For Pin we need to multiply all the intensities going in by area For Pout, we need to multiply all the intensities going out by area.
12. 12. 4 I rad (1 ) Tt 4 I abs Tt 4 I rad Ts 4 I rad Ts Pin = A(ασTt4+(1- α)σTt4)
13. 13. 4 I rad (1 ) Tt 4 I abs Tt 4 I rad Ts 4 I rad Ts Pin = A(ασTt4+σTt4- ασTt4)
14. 14. 4 I rad (1 ) Tt 4 I abs Tt 4 I rad Ts 4 I rad Ts Pin = A(ασTt4+σTt4- ασTt4) = AσTt4
15. 15. 4 I rad (1 ) Tt 4 I abs Tt 4 I rad Ts 4 I rad Ts Pout = A(2εσTs4+(1- α)σTt4)
16. 16. Pin = Pout AσTt4 = A(2εσTs4+(1- α)σTt4) ε=α, SO
17. 17. AσTt4 = A(2ασTs4+(1- α)σTt4)
18. 18. σTt4 = 2ασTs4+ σTt4 - ασTt4
19. 19. σTt4 = 2ασTs4+ σTt4 - ασTt4
20. 20. 0= 2ασTs4- ασTt4
21. 21. 0= 2Ts4- Tt4
22. 22. 0= 2Ts4- Tt4 +Tt4 +Tt4
23. 23. Tt4 = 2Ts4
24. 24. Tt Ts 1 2 4
25. 25. The temperature at which IR is escaping has to be 255 K because we have thermal equilibrium at this altitude with no absorption by greenhouse gases. Thus Tt = 255 K
26. 26. 255 K Ts 1 2 4
27. 27. And the answer is Ts = 214 K
28. 28. 214K 214K
29. 29. Now let us find the altitude at which Tt = 255 K T z
30. 30. o Tf Ti C 6 .7 z zi km f
31. 31. o C Tf Ti 6 .7 (z f zi ) km
32. 32. The average top of the troposphere is zf = 11 km, so o C 214 K Ti 6 .7 (11 km zi ) km
33. 33. o C 214 K 255 K 73 . 7 K 6 .7 zi km
34. 34. o C 41 K 73 . 7 K 6 .7 zi km +73.7K +73.7K
35. 35. K 32 . 7 K 6 .7 zi km
36. 36. 32 . 7 K zi K 6 .7 km
37. 37. And the answer is zi = 4.9 km
38. 38. HOMEWORK: 1. Use Tf = 214 K and zf = 11 km to find the surface temperature of the Earth with greenhouse gases. If CO2 doubles in the atmosphere, the top of the troposphere is predicted to rise to 11.6 km. What would the surface temperature of the Earth be then?