1. CLIMATE Climate Model
SOHO/Extreme Ultraviolet Imaging Telescope (EIT) consortium. Visual Tour of the Solar System: The Sun (online). About.com. [May 5, 2009].

2. PHYSICAL CLIMATE SYSTEM

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

4. Big Ideas
Climate Model
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).

5. Big Ideas 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

6. Big Ideas
Climate Model
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.

7. Black Body Radiation
Climate Model
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.

8. Black Body Radiation Climate Model
NASA. Electromagnetic spectrum (online). [May 19, 2009]
Figure 1. The electromagnetic spectrum with corresponding temperatures of radiation emitting bodies.

9. Black Body Radiation Methods of energy transfer by radiation:
Climate Model
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.

10. Black Body Radiation
Climate Model
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.

11. Stefan-Boltzmann Law
Climate Model
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 x10-8 W/m2∙K4 A 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)

12. How much power does the Sun radiate onto Earth?
Solar Radiation
Climate Model
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).

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

14. Solar Radiation
Climate Model
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.

15. Solar Radiation
Climate Model
Explain using a balloon and a coin.

16. Power Equations
Climate Model
Using the Stefan-Boltzmann Law, and assuming the Sun is a black body (ε = 1) Ps = 4πrs2σTs4 Ps ≈ 3.9x1026 W Thus, Earth’s incident solar power can be found as Pe = f ∙ Ps Pe ≈ 1.77x1017 W

17. Albedo
Climate Model
A fraction of solar radiation is reflected straight back into space without ever warming the Earth.
NASA/Goddard Space Flight Center, Scientific Visualization Studio. Apollo 17 30th Anniversary: Saudi Arabia (online). Nasa. [May 4, 2009].

18. Albedo This reflective property is called the albedo, A.
Climate Model
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 ≈ 1.23x1017 W

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

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

21. How much power does Earth radiate?
Power Equation
Climate Model
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.

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

23. Greenhouse Effect
Climate Model
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.

24. Greenhouse Effect
Climate Model
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.

25. Greenhouse Effect
Climate Model
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.

26. Greenhouse Effect 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.

27. Greenhouse Effect
Climate Model
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.

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

29. Greenhouse Effect 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, Iin or Iout , where ℰ = 1.

30. EARTH’S ATMOSPHERE SURFACE
Implications
Climate Model
The temperature implications of this model are as follows:
EARTH’S ATMOSPHERE SURFACE
EARTH’S SURFACE
Iin = 240 W/m2
Iin = 480 W/m2
Iin = Iout
= σTa4
= σTe4
Ta = 255 k = -18°C
Ta = 303 k = 30°C

31. Greenhouse Effect
Climate Model
This temperature for Earth’s surface is much too hot! Earth’s mean surface temperature is recorded as a mean of °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.

32. Greenhouse Effect
Climate Model
Greenhouse Effect

33. Greenhouse Effect
Climate Model

34. Emissivity
Climate Model
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.
Çengel, Yunus A. Steady Heat Conduction. In: Heat Transfer a Practical Approach (2). New York: McGraw Hill Professional, 2003, p. 173.

35. Emissivity
Climate Model
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

36. Emissivity Climate Model
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, Iin or Iout , where ℰ =0.9.

37. EARTH’S ATMOSPHERE SURFACE
Implications
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.
EARTH’S ATMOSPHERE SURFACE
EARTH’S SURFACE
Ta = K = -30.1°C
Ta = K = 15.8°C

38. Bibliography
SOHO/Extreme Ultraviolet Imaging Telescope (EIT) consortium. Visual Tour of the Solar System: The Sun (online). About.com. [May 5, 2009].
NASA. Electromagnetic spectrum (online). [May 19, 2009]
NASA/Goddard Space Flight Center, Scientific Visualization Studio. Apollo 17 30th Anniversary: Saudi Arabia (online). Nasa. [May 4, 2009].
Çengel, Yunus A. Steady Heat Conduction. In: Heat Transfer a Practical Approach (2). New York: McGraw Hill Professional, 2003, p. 173.

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

51. How to solve the equations

52. Parameterizations

54. TAKE HOME MESSAGE
There is considerable confidence that climate models provide credible quantitative estimates of future climate change, particularly at continental scales and above.
This confidence comes from the foundation of the models in accepted physical principles and from their ability to reproduce observed features of current climate and past climate changes.
Confidence in model estimates is higher for some climate variables (e.g., temperature) than for others (e.g., precipitation). Over several decades of development, models have consistently provided a robust and unambiguous picture of significant climate warming in response to increasing greenhouse gases