Introduction to Climate and
Energy Balance Models
July 22, 2013
Samantha Oestreicher
University of Minnesota
“Some say the world will end in fire…”
Outline
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What is Climate?
How do we observe climate?
An Overview of Earth’s Climate System.
Types of Radiation
Energy Balance Model
◦ Stefan-Boltzmann
◦ Budyko-Sellers
What is Climate?
Climate := 30 year average
of weather.
What is Climate?
Climate := 30 year average
of weather.
Weather:
Do I need an umbrella today?
What is Climate?
Climate := 30 year average
of weather.
Weather:
Do I need an umbrella today?
Climate:
Do I need to own an umbrella?
How do we observe climate?
People have been making observations
for hundreds of years.
How do we observe climate?
How do we observe climate?
http://spaceplace.nasa.gov/earth-card-game/terra-lrg.en.png
How do we observe climate?
http://www.dartmouth.edu/~mpayres/People/Sharon.7506.web.jpg
http://www.whoi.edu/ooi_cgsn/auvs-gliders?tid=1621&cid=137956&article=95673
How do we observe climate?
How do we observe climate?
nicl-smo.unh.edu
How do we observe climate?
nicl-smo.unh.edu
An Overview of Earth’s Climate System
9:35
How do we model climate?
http://www.prism.washington.edu/story/Earth+System+Models
How do we model climate?
There are two main view on how to model climate:
1. “No detail is too small!”
Leads to all-inclusive
Global Climate Models
www.pmel.noaa.gov/foci/ice06/FOCI_Ice2006_phytoplankton.html
2. “The rest is details”
Leads to simple
Conceptual
Climate Models
www.nasa.gov/vision/earth/lookingatearth/ice_clouds.html
Global Climate Models
Complicated
choices starting
from how to grid
the globe.
Global Climate Models
Global Climate Models
Global Climate Models require:
Physical sciences
• Physical, chemical, biological
processes
Computer science
• Data mining, coupling non-similar
grids, error analysis, parallel
processing, time optimization
Statistics
• Extreme events, trends, and
averaging
Mathematics
• Data assimilation, numerical
znalysis, PDEs
Global Climate Models - Simulation
Global Climate Models - Prediction
IPCC Report AR4
9:50
How do we model climate?
There are two main view on how to model climate:
1. “No detail is too small!”
Leads to all-inclusive
Global Climate Models
www.pmel.noaa.gov/foci/ice06/FOCI_Ice2006_phytoplankton.html
2. “The rest is details”
Leads to simple
Conceptual
Climate Models
www.nasa.gov/vision/earth/lookingatearth/ice_clouds.html
Energy Balance Models
Temperature Change = Energy In – Energy Out
Energy Out using Stefan-Boltzmann Law:
Temperature of the Sun = 5,778 K
Power flux (W/m2) = (5.67 x10-8 )*(5778)4 = 6.33x107 W/m2
Question:
What kind of energy is the Sun radiating?
Types of Radiation
http://www2.webster.edu/~barrettb/courses/mediaproduction.htm
Types of Radiation
Plank’s Function gives a distribution of wavelengths based on the temperature of
the body.
Wein’s Law tells us the maximum frequency is inversely proportional to the
temperature. ie: Hotter bodies produce shorter wavelengths.
The Sun gives off
shortwave radiation
or ultraviolet.
The Earth gives off
longwave radiation or
infared.
Types of Radiation
The Sun gives off
shortwave radiation
or ultraviolet.
The Earth gives off
longwave radiation or
infared.
http://www2.webster.edu/~barrettb/courses/mediaproduction.htm
Energy Balance Models
In the “Global Energy Balance Models and the Goldilocks Zone” section of the
MATLAB guide, you will use the Stefan-Boltzmann Law to derive the average
incoming solar radiation (or insolation) to Earth.
Earth’s Insolation = 342 W/m2 = Q
Thus the simplest energy balance model is:
Temperature change = energy in – energy out
R
dT
 Q  T
4
dt
Which has equilibrium solution:
or
Q = sTeq4
(342/s)1/4=Teq
Thus Earth’s temperature is modeled to be
Teq = 279K = 6 °C = 43 °F
Energy Balance Models
Teq = 279K = 6 °C = 43 °F
But the observed temperature of Earth is only
T = 14 °C
Stefan-Boltzmann is black body radiation. We need to include albedo.
Globally 30% of
insolation is
reflected back
into space.
http://www.cocorahs-albedo.org/
Energy Balance Models
Energy Balance Models
Thus the improved energy balance model is:
R
dT
 Q (1   )   T
4
dt
Which has equilibrium solution:
Q (1- a) = sTeq4
or
(342*(1-0.3)/s)1/4=Teq
Thus Earth’s temperature is modeled to be
Teq = 255K = -18 °C = 0 °F
Question:
Why isn’t the Earth a snowball?
Energy Balance Models
Budyko - Sellers Suggest new outgoing longwave radiation (OLR) formulation:
OLR = A + BT
A and B are determined from satellite observations.
T is surface temperature (in Celsius).
A = 202 W/m B =1.90 W/m K
Dynamics
photosphere
temperature
global mean
surface temperature
R
dT
 Q (1   )   T
4
dt
R
dT
 Q (1   )  ( A  BT )
dt
Question:
What is happening in the atmosphere to cause this discrepancy?
Energy Balance Models
Budyko-Sellers Energy Balance Model is:
R
dT
 Q (1   )  ( A  BT )
dt
With equilibrium solution
T eq 
Q (1   )  A
B
This equilibrium solution is stable with eigenvalue –B. (Recall B>0.)
Question:
What if Earth’s albedo was not 30%?
 ice  0 . 62
 ocean  0 . 32
Budyko 1969
Last Question:
Extras