CLIMATE Simple Earth Climate Model

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CLIMATESimple Earth Climate Model
Lecture Notes
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Goals
Climate Model

• To develop a simple model of Earths Climate.
• To develop a model of the greenhouse effect.

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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 Earths temperature is constant
  (14C).

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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.

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Black Body Radiation
Climate Model

• Black-body radiation an objects 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.

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

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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.

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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-Boltzmanns
  Law
• P sAeT4
• P is the power radiated, or the amount of
  energy per second (units
• Watts, W)s is the Stefan-Boltzmann
  constant, equal to 5.6696x10-8 W/m2K4A is
  the area of emission (units square metres, m2)e
  is the emissivity of the object, or the
  fraction of EM radiation a
• surface absorbs (0 e 1)T is the
  temperature of the object (units Kelvins, K)

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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).

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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.

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Solar Radiation
Climate Model
thus, pinhead, basketball, 29.2 metres between
Image 1. A scale image of the Earth in relation
to the Sun. The Earth is represented by the
white pinhead and the Sun by a basketball. The
two are 29.2 m apart, approximately the length of
a basketball court.
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Solar Radiation
Climate Model

• Image 2A. A scale image of the Earth represented
  by a white pinhead.

Image 2B. A scale image of the Sun represented
by a basketball.
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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.

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Solar Radiation
Climate Model

• Figure 3. The projected area of Earth on the
  spherically distributed solar radiation emitted
  by the Sun. Above, Earth is a disk with a
  radius, re, of 6.37x106 km. The dashed lines
  indicate the slice of incident solar radiation on
  Earth. Image not to scale.

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Power Equations
Climate Model

• Using the Stefan-Boltzmann Law, and assuming the
  Sun is a black body (e 1)
• Ps 4prs2sTs4 Ps 3.9x1026 W
• Thus, Earths incident solar power can be found
  as
• Pe f Ps Pe 1.77x1017 W

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Albedo
Climate Model

• A fraction of solar radiation is reflected
  straight back into space without ever warming the
  Earth.

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Albedo
Climate Model

• This reflective property is called the albedo, A.
• For Earth, A0.3, and is mainly due to clouds,
  haze and ice.
• Therefore, Earths incident power must have a
  correction term, where
• Pin (1 A) Pe
• Pin 1.23x1017 W

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Solar Intensity
Climate Model

• The incident solar radiation, S, on the surface
  of Earths atmosphere that the sunlight shines on
  is

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Solar Intensity
Climate Model

• The mean incident solar intensity, Iin , on the
  entire surface of Earth as averaged over the
  entire year is

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Power Equation
Climate Model

• How much power does Earth radiate?
• The power emitted by Earth is
• Pout 4pre2sTe4
• where the Earth is assumed to be a black body, so
  e 1.

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Solar Intensity
Climate Model

• The solar intensity emitted from Earths surface
  is

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Greenhouse Effect
Climate Model

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

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Greenhouse Effect
Climate Model

• Earths 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
  Earths radiation from exiting.
• This is called the Greenhouse Effect because
  glass behaves in a similar fashion.

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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.

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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.
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Greenhouse Effect
Climate Model

• To incorporate the greenhouse effect into our
  simple model lets make the following
  assumptions
• there is only one layer of Earths 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.

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Greenhouse Effect
Climate Model

• The equation for the conservation of energy on
  Earths surface is
• The equation for the conservation of energy of
  Earths atmosphere becomes

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

• The temperature implications of this model are as
  follows

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Greenhouse Effect
Climate Model

• This temperature for Earths surface is much too
  hot! Earths mean surface temperature is
  recorded as a mean of 14.5C.
• 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.

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Greenhouse Effect
Greenhouse Effect
Climate Model
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Greenhouse Effect
Climate Model
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Emissivity
Climate Model

• To improve our model, we will focus on the first
  of these factors.
• There are holes in our atmosphere, so Earths
  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 Earths surface as well as
  Earths atmosphere.

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Emissivity
Climate Model

• The equation for the conservation of energy on
  Earths surface is now
• The equation for the conservation of energy of
  Earths atmosphere becomes

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Emissivity
Climate Model

• Figure 5. A diagram of the exchange of EM
  radiation between the Sun, Earth, and Earths
  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.

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Implications
Climate Model

• From this data, the temperature implications are
  as follows
• Therefore, this corrected model produces a mean
  temperature for Earths surface that is very
  close to the measured mean temperature of 14.5C.
  

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Bibliography
Climate Model

• SOHO/Extreme Ultraviolet Imaging Telescope (EIT)
  consortium. Visual Tour of the Solar System The
  Sun (online). About.com. http//space.about.com/o
  d/solarsystem/ss/visualtourss.htm May 5, 2009.
• NASA. Electromagnetic spectrum (online).
  http//mynasadata.larc.nasa.gov/glossary.php?word
  electromagnetic20spectrum 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/a0026
  81/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.

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