CLIMATE Climate Model

1. CLIMATEClimate Model

2. PHYSICAL CLIMATE SYSTEM

3. To develop a simple model of Earth’s Climate. • To develop a model of the greenhouse effect. Climate Model Goals

4. Model: a simplified and idealized physical and mathematical construct that allows one to understand and make useful predictions about a real system. Steady-state: mean power coming in (Pin) must equal the mean power going out (Pout), all the time. Thus Earth’s temperature is constant (~14°C). Climate Model Big Ideas

5. Model Partial Differential Equations • Conservation of momentum • Conservation of mass • Conservation of water • Conservation of certain chemical species • First law of Thermodynamics • Equation of state • Radiative transfer equations Big Ideas

6. The Earth is a closed thermodynamic system, freely exchanging energy with the rest of the universe, but not matter (except for tiny amounts). The Earth is a vacuum thus energy is lost in the form of radiation. Climate Model Big Ideas

7. Black-body radiation: an object’s temperature determines at what rate radiation is emitted, and at what wavelengths. A black body is an idealized object that is a perfect absorber as well as a perfect emitter of electromagnetic (EM) radiation. Climate Model Black Body Radiation

8. Climate Model Black Body Radiation Figure 1. The electromagnetic spectrum with corresponding temperatures of radiation emitting bodies.

9. Methods of energy transfer by radiation: • Transmission: It can pass through the object. ie. A window. • Reflection: emission from a surface. ie. A mirror. • Absorption: The radiation is retained within the object it hits. The object will then emit energy as black body radiation depending on its temperature. Climate Model Black Body Radiation

10. The wavelength of emitted radiation depends on the temperature of the black body object. • The temperature of a black body depends on the percentage of radiation that is absorbed and re-emitted. Climate Model Black Body Radiation

11. The energy of EM radiation that is emitted or absorbed by an object depends mainly on its temperature, as shown by the Stefan-Boltzmann’s Law: P =σAεT4 P is the power radiated, or the amount of energy per second (units: Watts, W)σ is the Stefan-Boltzmann constant, equal to 5.6696x10-8 W/m2∙K4A is the area of emission (units: square metres, m2)ε is the emissivity of the object, or the fraction of EM radiation a surface absorbs (0≤ ε ≤1)T is the temperature of the object (units: Kelvins, K) Climate Model Stefan-Boltzmann Law

12. How much power does the Sun radiate onto Earth? • Sunlight, or solar radiation, includes the total spectrum of electromagnetic radiation given off by the Sun. • This solar radiation is emitted in a spherical distribution. • No solar power is absorbed by interplanetary space (a vacuum). Climate Model Solar Radiation

13. Figure 2. The solar radiation, emitted by the Sun in a spherically symmetric distribution, coming into contact with Earth. Image not to scale. Climate Model Solar Radiation

14. The relative size of Earth is incredibly tiny in relation to the Sun • It can be approximated that the ratio of its projected 2D area on the 3D surface area of the solar radiation distribution is equal to the fraction, f, of the solar power incident on the Earth. Climate Model Solar Radiation

15. Climate Model Solar Radiation

16. Using the Stefan-Boltzmann Law, and assuming the Sun is a black body (ε = 1) Ps= 4πrs2σTs4Ps≈ 3.9x1026W Thus, Earth’s incident solar power can be found as Pe=f ∙ PsPe≈ 1.77x1017W Climate Model Power Equations

17. A fraction of solar radiation is reflected straight back into space without ever warming the Earth. Climate Model Albedo

18. This reflective property is called the albedo, A. • For Earth, A≈0.3, and is mainly due to clouds, haze and ice. • Therefore, Earth’s incident power must have a correction term, where Pin= (1 – A) ∙ Pe Pin≈ 1.23x1017W Climate Model Albedo

19. The incident solar radiation, S, on the surface of Earth’s atmosphere that the sunlight shines on is Climate Model Solar Intensity

20. The mean incident solar intensity, Iin , on the entire surface of Earth as averaged over the entire year is: Climate Model Solar Intensity

21. How much power does Earth radiate? • The power emitted by Earth is Pout= 4πre2σTe4 where the Earth is assumed to be a black body, so ε = 1. Climate Model Power Equation

22. The solar intensity emitted from Earth’s surface is Climate Model Solar Intensity

23. The simple model so far assumes that Earth lacks an atmosphere. • Earth’s atmosphere is mostly transparent to solar radiation (44% visible, 52% near infrared (IR), 4% ultraviolet (UV)). • Therefore, most of Earth’s incident solar radiation comes through the atmosphere and warms us. Climate Model Greenhouse Effect

24. Earth’s atmosphere also absorbs much of its own radiation (longer wavelength IR). • The atmosphere acts like one way glass, allowing solar radiation to enter, but preventing the Earth’s radiation from exiting. • This is called the Greenhouse Effect because glass behaves in a similar fashion. Climate Model Greenhouse Effect

25. Did you know... We can see through windows because our eyes absorb visible light. If, however, we were looking through infrared lenses, a window would appear to be a mirror. Climate Model Greenhouse Effect

26. Climate Model Image 3. The image on the left is taken with a regular camera and illustrates the properties of visible light. The image on the right is taken with an infrared camera and shows the windows emitting infrared radiation (in the form of hear) and illustrate that they are no longer appear transparent. Greenhouse Effect

27. To incorporate the greenhouse effect into our simple model let’s make the following assumptions: • there is only one layer of Earth’s atmosphere. • the atmosphere allows most of the incident solar radiation through, but absorbs radiation emitted by Earth. • the atmosphere then radiates equally from both its topside and underside. Climate Model Greenhouse Effect

28. The equation for the conservation of energy on Earth’s surface is The equation for the conservation of energy of Earth’s atmosphere becomes Climate Model Greenhouse Effect

29. Climate Model Figure 4. A diagram of the exchange of EM radiation between the Sun, Earth, and Earth’s atmosphere. The green arrows represent the incident solar intensity, which is not absorbed by Earth’s atmosphere. The red arrows represent IR radiation. The red equations represent the mean solar intensity, Iinor Iout , where ℰ = 1. Greenhouse Effect

30. The temperature implications of this model are as follows: Climate Model Implications

31. This temperature for Earth’s surface is much too hot! Earth’s mean surface temperature is recorded as a mean of 14.5°C. • This model assumes a single but perfect greenhouse layer, which in reality is not accurate. • In reality, there are many factors that contribute to this difference. Climate Model Greenhouse Effect

32. Greenhouse Effect Climate Model Greenhouse Effect

33. Greenhouse Effect Climate Model

34. To improve our model, we will focus on the first of these factors. • There are holes in our atmosphere, so Earth’s atmosphere only absorbs a fraction of the IR radiation that Earth emits. • In other words, ℰ ≠ 1, but ℰ = 0.9, the emissivity of air. • Therefore, an observer in space would detect IR radiation emitted by Earth’s surface as well as Earth’s atmosphere. Climate Model Emissivity

35. The equation for the conservation of energy on Earth’s surface is now The equation for the conservation of energy of Earth’s atmosphere becomes Climate Model Emissivity

36. Figure 5. A diagram of the exchange of EM radiation between the Sun, Earth, and Earth’s atmosphere. The green arrows represent the incident solar intensity. The red arrows represent IR radiation. The red equations represent the mean solar intensity, Iinor Iout , where ℰ =0.9. Climate Model Emissivity

37. From this data, the temperature implications are as follows: Therefore, this corrected model produces a mean temperature for Earth’s surface that is very close to the measured mean temperature of 14.5°C. Implications

38. SOHO/Extreme Ultraviolet Imaging Telescope (EIT) consortium. Visual Tour of the Solar System: The Sun (online). About.com. http://space.about.com/od/solarsystem/ss/visualtourss.htm[May 5, 2009]. • NASA. Electromagnetic spectrum (online). http://mynasadata.larc.nasa.gov/glossary.php?&word=electromagnetic%20spectrum [May 19, 2009] • NASA/Goddard Space Flight Center, Scientific Visualization Studio. Apollo 17 30th Anniversary: Saudi Arabia (online). Nasa. http://svs.gsfc.nasa.gov/vis/a000000/a002600/a002681/index.html [May 4, 2009]. • Çengel, Yunus A. Steady Heat Conduction. In: Heat Transfer a Practical Approach (2). New York: McGraw Hill Professional, 2003, p. 173. Bibliography

39. What is a CLIMATE MODEL? • Designing a model • Spatial grid • Continuity equation • Time Step and stability • Solving the equation • Reality…computation time and parameterization

40. A model that incorporates the principles of • Physics, chemistry, biology into a • mathematical model of climate • e.g. GCM (Global Circulation Model) • Such a model has to answer what happens to • temperature, precipitation, humidity, wind • speed and direction, clouds, ice and other • variables all around the globe over time

41. IPPC Report

44. Schematics representation climate model MIT-IGSM

45. Spatial grid Divide the Earth’s atmosphere Into a finite number of boxed Consider that each variable has the same value throughout the cell Write a budget for each cell, defining the change within the box and the flow between the cells

47. Continuity Equations Change in grid cell can be expressed at given time step

48. Continuity Equations (Atmosphere/Oceans in 3-D)

49. Equations for Chemistry and Biology

50. Time stepping and stability