Equilibrium Temperature
          with
   Greenhouse Gases
T
            z
Equation for temperature change
with altitude in the troposphere
Γ is called the lapse rate for the
tropos...
o
                      C
           6 .7
                  km
Average lapse rate for dry air
in the troposphere
At this altitude, 95% of IR escapes to space without
further absorption.
We want to find what the altitude is for
95% escape of IR,



And the surface temperature of the Earth
with greenhouse gas...
We will use a two system model of the
atmosphere. The two systems being
the troposphere and the stratosphere.



We will a...
Stratosphere


Troposphere
4
I abs   Tt
4
        I rad    (1   ) Tt


             4
I abs   Tt
4
        I rad            (1   ) Tt


                     4
I abs           Tt                                          ...
For thermal equilibrium,
        power in = power out
               Pin=Pout


For Pin we need to multiply all the
intens...
4
        I rad            (1   ) Tt


                     4
I abs           Tt                                          ...
4
        I rad            (1   ) Tt


                     4
I abs           Tt                                          ...
4
        I rad            (1   ) Tt


                     4
I abs           Tt                                          ...
4
        I rad            (1   ) Tt


                     4
I abs           Tt                                          ...
Pin = Pout

AσTt4 = A(2εσTs4+(1- α)σTt4)


           ε=α, SO
AσTt4 = A(2ασTs4+(1- α)σTt4)
σTt4 = 2ασTs4+ σTt4 - ασTt4
σTt4 = 2ασTs4+ σTt4 - ασTt4
0= 2ασTs4- ασTt4
0= 2Ts4- Tt4
0= 2Ts4- Tt4
+Tt4    +Tt4
Tt4 = 2Ts4
Tt
Ts        1
     2    4
The temperature at which IR is
escaping has to be 255 K because we
have thermal equilibrium at this
altitude with no absor...
255 K
Ts        1
      2   4
And the answer is



  Ts = 214 K
214K
214K
Now let us find the altitude at which
                Tt = 255 K



             T
              z
o
Tf      Ti              C
             6 .7
z       zi          km
    f
o
                     C
Tf   Ti   6 .7           (z f   zi )
                 km
The average top of the troposphere
  is zf = 11 km, so


                         o
                             C
214 K  ...
o
                                      C
214 K   255 K   73 . 7 K   6 .7           zi
                                  km
o
                               C
  41 K   73 . 7 K   6 .7           zi
                           km
+73.7K   +73.7K
K
32 . 7 K   6 .7        zi
                  km
32 . 7 K
zi
            K
     6 .7
            km
And the answer is



 zi = 4.9 km
HOMEWORK:
1. Use Tf = 214 K and zf = 11 km to
find the surface temperature of the
Earth with greenhouse gases.


If CO2 do...
Equilibrium Temp W Ghg
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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?

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