1. Very Basic Climate Modeling Spring 2012, Lecture 5 1

2. Radiation Balance The temperature of any body, including earth, is determined by a balance between incoming radiation and outgoing radiation This is similar to a bank account F in = F out  F in is the incoming energy flux  F out is the outgoing energy flux 2

3. Intensity of Incoming Energy Sunlight, at the earth’s distance from the sun, has an intensity, I, given by: I in = 1350 w/m 2 Some of the incoming radiation is reflected back into space – reflectivity is called the albedo, denoted α (Greek alpha) For earth, average albedo is about 0.33 3

4. Corrected I in Correcting for reflection, we get I in = 1350 w/m 2 (1 – α) = 1000 w/m 2 We must correct this for the area of the earth which receives solar illumination Only half of the earth is at any given time, and sunlight near the poles is much weaker than near the equator 4

5. Flux in Watts In order to measure the flux in watts, rather than watts/m 2, we need to multiply by the area The illuminated area is actually that of a circle A[m 2 ] = πr 2 earth (Greek pi) So the incoming flux in watts becomes  F in = πr 2 earth (1-α) I in 5

6. Computing F out Next we need to compute the outbound energy flux, F out In order to do this, we use the Stefan- Boltzmann equation for a blackbody radiator 6

7. Stefan-Boltzmann Equation I = εσT 4  I is the Intensity of emitted radiation  ε is the emissivity (Greek epsilon) o ε is a number between 0 and 1 o If a blackbody is perfect, ε = 1  σ is a fundamental constant of physics called the Stefan-Boltzmann constant (Greek sigma)  T is absolute temperature (K) Note that intensity varies as the fourth power of the absolute temperature 7

8. F out 8 The earth radiates over the entire surface The area of the earth is given by A = 4πr 2 earth F out is computed by F out = AεσT 4 earth = 4πr 2 earth εσT 4 earth

9. Equating Incoming and Outgoing Flux Substituting in F in = F out gives πr 2 earth (1-α) I in = 4πr 2 earth εσT 4 earth Eliminating factors common to both sides gives: (1-α) I in = 4εσT 4 earth We know everything except T, so rearrange T 4 earth = [(1-α) I in ]/[4εσ] 9

10. T earth Solving for T: T earth = 4 √[(1-α) I in ]/[4εσ] 10

11. Incoming vs. Outgoing Flux Diagram 11

12. Model Construction What we have done so far is to construct what scientists call a “model” Models attempt to mimic the behavior of the natural system we are studying Models can be constructed as computer programs, which allow scientists to change conditions, and to attempt to assess what effects a given change, such as a doubling of atmospheric carbon dioxide, will cause 12

13. Early Climate Models The earliest climate models were physical – a dishpan full of water on a record turntable, heated on the rim by a flame representing the sun Such models were made to examine circulation patterns in the atmosphere, and were called Global Circulation Models Today this same name is used for far more sophisticated models 13

14. Model Prediction If we calculate T earth, we get a value of about 255 K, or about -18 ◦ C We have seen before this is too cold Thus, we learn a very important lesson about models – they are only as good as the mathematical representation of the real world, and the data that we put in Modelers have an acronym – GIGO – garbage in, garbage out 14

15. Are Models Useful? Does this mean mathematical modeling is a waste of time?  No, because models do allow us to understand certain phenomena, such as water vapor and sea ice feedbacks fairly well  They are relatively inexpensive, and often suggest where new avenues of research are needed (clouds, aerosol particles)  However, they must be used with caution, and the results understood in the context of what the model was written to do 15

16. What Happens When a Model is Wrong? Observation: Our model result for worldwide temperature is much too low Action  Throw out the model?  Improve the model? o Add more features to it so it better mimics the real world o Attempt to collect better, or new, data so the model can be improved 16

17. Improving Our Model Our model is known to predict a temperature much lower than the observed global temperature We also know that the earth has a greenhouse effect, and this is not in our model We need to add atmospheric layers to allow for the greenhouse effect 17

18. Adding a Layer Our new model uses a hypothetical “pane of glass” suspended above the earth’s surface It is transparent to visible light It acts like a blackbody for infrared light, absorbing all of it, and reradiating it in all directions 18

19. Model Changes Incoming radiation is unaffected Outgoing radiation must be changed  Instead of I out, we now have I up, earth  When infrared radiation strikes the glass pane, it is absorbed and reradiated o Half goes down, as I down, atmosphere o The other half goes up, as I up, atmosphere 19

20. Atmospheric Energy Budget We can write an atmospheric energy budget  I up, earth = I up, atmosphere + I down, atmosphere Alternatively,  2εσT 4 atmosphere = εσT 4 earth 20

21. Ground Energy Budget I up, earth = I in, solar + I down, atmosphere εσT 4 earth = (1-α)/4*I solar + εσT 4 atmosphere 21

22. Whole Earth Energy Budget 22 The whole earth budget can be represented by assuming that what goes in must come out, so that: I in, solar = I up, atmosphere

23. Solving the System εσT 4 atmosphere = (1-α)/4*I solar We now have two unknowns, T earth and T atmosphere This equation has the same form as the one for the bare earth  T earth = 4 √[(1-α) I in ]/[4εσ] This is important! It means that the temperature is controlled by the place at which earth radiates into space We can call this the “skin” temperature, T skin 23

24. Skin Temperature We now know that T atmosphere = T skin Plugging  εσT 4 atmosphere = (1-α)/4*I solar Into  εσT 4 earth = (1-α)/4*I solar + εσT 4 atmosphere Gives  εσT 4 earth = 2εσT 4 atmosphere 24

25. Solving the Model T ground = 4 √2 T atmosphere The fourth root of 2 is 1.189 Thus, the ground temperature of earth is about 19% warmer than it would be without the greenhouse layer We got a value of 255 K from the bare earth model – 1.189*255 = 303 K, which is a bit too warm – actual global average temperature is about 288, or 15ºC 25

26. Why is the Model Incorrect? - 1 Like all models, this one does not fully reproduce the natural system – in fact, it is very simplified We have treated all wavelengths of radiation the same, but we know that there are “windows” through which some IR radiation is barely absorbed, while other wavelengths are fully absorbed 26

27. Why is the Model Incorrect? - 2 There is nothing in the model that reflects to the concentration of GHG It does not allow for clouds, or for different degrees of surface reflectivity It does not allow the presence of aerosol particles Model assumes heat transfer from ground to atmosphere is by radiation alone, ignoring convection And many more…. 27

28. Improving the Model 28 We can further improve the model by adding more layers, such as the two layer model seen here Layers might represent troposphere and stratosphere, for example Such models become more complex, but are still soluble

29. Archer Sink Model 29

30. Partially Plugged Sink 30 The partial blockage of the sink raises water levels, until the increased pressure drains the sink as fast as water flows in – analogous to temperatures increasing in the atmosphere