geography worksheet 2
Cooper2021
geographyweek6material1.pdf
Planetary Energy Budget
Through this week’s lectures, we will learn about the planetary energy budget and use this knowledge to understand a climate simulation model. This model will be key to developing a better understanding about our future.
As we learned in previous lectures, this figure demonstrates the incoming and outgoing energy budget.
Stefan-Boltzman Law • Energy intensity radiated by a perfect radiator
(blackbody) is: E=sT4 – Where E is a combination of intensity and
emissivity; represents the total rate of energy emission from the object at all frequencies
– Where sigma (s) is a fundamental constant of physics that never changes, also known as the Stefan-Boltzmann constant: s = 5.67 x 10-8 W m-2 K-4
– Where T is the temperature in Kelvins
We have learned that infrared light emission is associated with a blackbody, and its spectrum depends on the temperature of the object. There is an equation that allows us to identify how quickly energy is radiated from a blackbody object. This is called the Stefan-Boltzmann equation.
T, the temperature in Kelvins, is expressed as the superscript 4, which is an exponent. The Kelvin temperature scale begins with 0 K at which point atoms vibrate/move as little as possible; a temperature called absolute zero. Please note that there are no negative temperatures on the Kelvin scale.
So what does this Stefan-Boltzman equation tells us? -- A hot object emits much more light than a cold object!
Simple Climate Model
• Energy coming in from Sun must Equal Energy radiating back into space: in=out.
S/4(1-Albedo) = radiation out to space S/4(1-Albedo) = sT4
T = (S/4(1-Albedo)/s)1/4
In climate science, “models” are used in two different ways. One way is to make forecasts. For this purpose, a model should be as realistic as possible and should capture or include all processes that might be relevant in nature – typically, mathematical models are implemented on a computer. Once such a model has been constructed, a climate scientist can perform “what-if” experiments on it that could never be done in the real world, to determine how sensitive the climate would be to changes in the brightness of the sun or properties of the atmosphere, for instance.
We can also create a simple climate model by ourselves. Here, let’s try to construct a simple model called a “Layer Model”. This simple model is not intended to make detailed forecasts of our future climate, rather, it is used to better understand the workings of the real climate system.
In a Layer Model, outgoing energy flux equals incoming energy flux (in = out). Thus, a Layer Model is exactly in balance.
We will learn about the equations listed here, as well as the Layer Modal, in a little bit. For now, please know that albedo, which is an important component of this model, is defined as a “percentage of incoming radiation reflected back to space”. Thus, albedo is a measure of the reflectivity of the Earth’s surface. Ice, with white snow on top of it, has a high albedo: most sunlight hitting the surface reflects back towards space. Water and brown dirt, for instance, are much more absorbent and less reflective and has a lower albedo than ice.
�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)
ΔF (Wm-2) = α ln(C/C0) α = 5.35 Then, ΔT(K) = λ*ΔF
Svante Arrhenius Climate Sensitivity: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.
IPCC estimate of climate sensitivity 3.8�C � 0.78�C in the SAR 1995 (17 models) 3.5�C � 0.92�C in the TAR 2001 (15 models) 3.26�C � 0.69�C AR4 2007 (18 models)
Range is ~1.5 to 4.5ºC per doubling of CO2 �sensitivity above 4.5 ºC cannot be ruled out�
Arrhenius’s greenhouse law
Svante Arrhenius, a Swedish scientist, received a Nobel Prize for Chemistry in 1903.
Arrhenius developed a theory to explain the ice ages, and he attempted to calculate how
changes in the levels of carbon dioxide in the atmosphere could alter the surface
temperature through the greenhouse effect. Importantly, Arrhenius formulated his
greenhouse law. You can find his greenhouse law, which is the quoted statement in this slide, “if the quantity…”.
His greenhouse law formula remains important at present, and is heavily used. The
greenhouse law formula is:
ΔF = α ln(C/C0) Where C is carbon dioxide (CO2) concentration measured in parts per million by volume
(ppmv); C0 denotes an initial (or reference) CO2 concentration, and ΔF is the change in
the amount of energy reaching the Earth's surface (the radiative forcing) measured in
watts per square meter. Constant α is 5.35. Ln denotes a natural logarithm.
(continue)
�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)
ΔF (Wm-2) = α ln(C/C0) α = 5.35 Then, ΔT(K) = λ*ΔF
Svante Arrhenius Climate Sensitivity: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.
IPCC estimate of climate sensitivity 3.8�C � 0.78�C in the SAR 1995 (17 models) 3.5�C � 0.92�C in the TAR 2001 (15 models) 3.26�C � 0.69�C AR4 2007 (18 models)
Range is ~1.5 to 4.5ºC per doubling of CO2 �sensitivity above 4.5 ºC cannot be ruled out�
Arrhenius’s greenhouse law
The equation indicates that the radiative forcing is proportionate to the increase/decrease in CO2 concentration through time.
Now that we know how to calculate the radiative forcing associated with an increase/decrease in CO2, how do we determine the associated temperature change? In order to project temperature change from the radiative forcing, we need to understand the concept of climate sensitivity.
As described, climate sensitivity is an estimate of how sensitive the climate is to an increase in radiative forcing. The climate sensitivity value tells us how much the planet will warm or cool in response to a given radiative forcing change. Temperature change is proportional to the change in the amount of energy reaching the Earth's surface (the radiative forcing), and the climate sensitivity is the coefficient of proportionality: ΔT = λ*ΔF Where ΔT is the change in the Earth's average surface temperature, λ is the climate sensitivity in Kelvin per Watts per square meter (K/[W/m2]), and ΔF is the radiative forcing. To calculate the change in temperature, we just need to know the climate sensitivity.
ΔF (Wm-2) = α ln(C/C0) α = 5.35 Then, ΔT(K) = λ*ΔF
Svante Arrhenius �Climate Sensitivity�: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.
IPCC (Intergovernmental Panel on Climate Change) estimates climate sensitivity as λ (i.e. how global temperature change responds to changes in climate forcing), and that value (λ) has been modified as we learn more about the variables in our climate system which improves the climate model.
Arrhenius’s greenhouse law
Below is further reading to aid your understand about climate sensitivity: https://www.carbonbrief.org/explainer-how-scientists-estimate-climate-sensitivity
The Greenhouse Effect
σTs 4 =
S 4 1− Albedo( )+σTe
4
σTs 4 = 2σTe
4
σTe 4 =
S 4 1− Albedo( )
Ts = 2 1 4Te
For the surface (1)
For the atmosphere (2)
(1)
(2)
(3)
(4)
Solar flux
Now we are ready to talk about the Layer Model. Suppose we treat the atmosphere as a single layer of gas and that this gas absorbs and re-emits all of the infrared radiation incident on it. Let us assume that it absorbs and emits infrared radiation equally well at all wavelengths, so that we can treat it as a blackbody, and that it has an albedo A in the visible spectrum, just like that of the real Earth.
What are the temperatures of the gas layer and of the surface beneath it? Let’s call the layer temperature Te and the surface temperature Ts.
Let the amount of sunlight striking the planet be equal to S/4 (the globally averaged solar flux) – See figure. The surface absorbs an amount of sunlight striking the planet equal to S/4 x (1-A), along with a flux of downward infrared radiation from the atmosphere equal to sigma*Te4. The atmosphere absorbs an amount of upward infrared radiation from the ground equal to sigma*Ts4, and it emits infrared radiation in both the upward and downward directions at a rate of sigma*Te4 (The real atmosphere also absorbs some of the incoming solar radiation, but we ignore that complication here).
The Greenhouse Effect
σTs 4 =
S 4 1− Albedo( )+σTe
4
σTs 4 = 2σTe
4
σTe 4 =
S 4 1− Albedo( )
Ts = 2 1 4Te
For the surface (1)
For the atmosphere (2)
(1)
(2)
(3)
(4)
Solar flux
Using these energy components, we can write the overall energy balance in the form of two equations.
For the surface (1) and for the atmosphere (2)
The (2) equation arises because the atmosphere radiates in both the upward and downward directions (what goes in, what comes out!). If we now substitute the (2) equation into the left-hand side of the (1) equation and subtract sigma*Te4 from both sides, we obtain (3), which is known as the Earth’s energy-balance formula. Dividing the (3) equation by sigma and then taking the fourth root of both sides yields an additional result (4).
The Greenhouse Effect
σTs 4 =
S 4 1− Albedo( )+σTe
4
σTs 4 = 2σTe
4
σTe 4 =
S 4 1− Albedo( )
Ts = 2 1 4Te
For the surface (1)
For the atmosphere (2)
(1)
(2)
(3)
(4)
Solar flux
What does (4) equation tells us? The surface temperature Ts is higher than the one-layer-atmosphere temperature Te by a factor of the fourth root of 2, or about 1.19. For Te = 255 K, as on Earth at present, we get Ts = 303 K, so we can calculate a greenhouse effect as… deltaTg = Ts – Te = 48 K Where Tg denotes temperature changes due to greenhouse effect This 48 K value is actually higher than the real greenhouse effect on Earth by about 15 K – so this simple model is not accurate. In the real world, some of the infrared radiation leaks through to space. Regardless, here are the simple basics of the climate model, and you have created one!
The atmosphere: • is transparent to incoming solar radiation
(short wavelengths in UV and visible part of electromagnetic spectrum)
• is opaque to outgoing longwave radiation in the infrared part of the spectrum
Take-home point (One-layer atmosphere model)
This is the take-home point about the layer model. Please know that in this particular simplified model, the incoming solar radiation does not cause interference, scatter, or get absorbed within the atmosphere. This is unrealistic and contrary to what happens in reality.
How much energy does the Earth receive from the Sun
• The sun has a luminosity (brightness) of ~3.86 X 1026 W
• It is ~150,000,000 km away (150x109 meters)
We earlier talked about the energy budget… how much energy do we receive anyway? Here are some important facts.
The solar constant • So, in our favorite units of Watts per square meter [Wm-2],
the energy shining on the top of the atmosphere So (solar constant) is:
• Where W is in Watts, m is meters. Remember the area of a sphere is 4 π r2 r (radius of the sphere) in this case is the Sun-Earth distance.
The answer is So = 1365 Wm-2
oS = 3.86×1026W
4•π •(150×109m)2
Average incoming solar energy
Because solar energy is not the same over the entire Earth at any given moment, we came to the conclusion that the average energy flux hitting the top of the atmosphere is exactly one fourth the solar constant (So/4) which equals
341 Wm-2
Planetary Albedo is ~0.3
We need to take albedo into consideration. Earth’s planetary albedo, which is defined as the ratio of reflected radiation from the surface to incident radiation upon it, is 0.3 or 30 %.
Accounting for albedo
• About 30% of the incoming energy is reflected by clouds and light colored things on the earth’s surface like snow and ice.
• So, the actual available energy is only 70% of 341 Wm-2 which equals 240 Wm-2. That’s the actual flux of energy that the Earth feels on average, at the top of the Atmosphere.
Simple Climate Model (without greenhouse effect)
• Energy coming in from the Sun must Equal Energy radiating back into space or “what comes in must equal what’s going out”
From the previous slide: So/4(1-Albedo)=radiation out to space
or
240 Wm-2 = radiation out to space
Let’s account for albedo in a simple climate model (not a Layer Model yet – here we don’t consider greenhouse effect).
Stefan-Boltzmann Law
• Energy intensity radiated by a perfect radiator (blackbody) is: E=sT4
– Where s is a constant = 5.67 x 10-8 W m-2 K-4
– Units of E are in Wm-2
Josef Stefan Ludwig Boltzmann
• So… according to Stefan-Boltzmann, we can calculate the theoretical temperature of Earth since we know how much energy is coming in from the sun (which is the same amount that’s leaving as long wave IR radiation)
Incoming Energy = sT4 (T is Earth’s temperature) 240 Wm-2=sT4
– Where s = 5.67 x 10-8 W m-2 K-4
Now we can solve for the Earth’s temperature T
T 4 = 240Wm2
σ or
T = 240Wm2
σ
!
" #
$
% &
1 4 = 255K 255K is only -18ºC! Brrrrrr.
255K is cold!
• Why is the number so off from the realistic mean Earth surface temperature?
• Because these calculations don’t account for the atmosphere or greenhouse effect, so we need a better climate model.
Account for the Greenhouse Effect (one layer atmosphere model)
€
σTs 4 =
S 4 1− Albedo( ) + σTe
4
σTs 4 = 2σTe
4
σTe 4 =
S 4 1− Albedo( )
Ts = 2 1 4 Te240 Wm-2 coming into surface
240 Wm-2 leaving into space
340 Wm-2 available before accounting for albedo
*Remember S is the solar constant 1365Wm2
Ts is earth surface temp. Te is temp at the top of the atmosphere where energy is radiated back to space
σTs 4 =
S 4 1− Albedo( )+σTe
4
σTs 4 = 2σTe
4
σTe 4 =
S 4 1− Albedo( )
Ts = 2 1 4Te
The Earth surface temp Ts is warmer than at the top of the atmosphere Te, because energy radiated from the surface is absorbed by the atmospheric greenhouse gasses and re-emitted back out to space AND back down to the earth’s surface. The atmosphere radiates both UP AND DOWN which is why
σTs 4 = 2σTe
4
What’s the point?
σTs 4 =
S 4 1− Albedo( )+σTe
4
σTs 4 = 2σTe
4
σTe 4 =
S 4 1− Albedo( )
Ts = 2 1 4Te
We already calculated earth’s temperature Te (255K). If we plug in 255K to the one layer model, we get a surface earth temperature of 303K. That’s a lot better than our last answer of 255K, but warmer than the actual number of 288K
What’s the point?
Ts = 2 1 4Te = 2
1 4255K =303K
Things to think about when you review today’s lecture
Here are a couple of suggestions if you would like to go further and explore more about this model.
• What would the temperature of the Earth be, if it had no atmosphere or greenhouse effect?
• What would the temperature of the Earth be, if the atmosphere was totally transparent to incoming solar radiation, and totally opaque to escaping long wave radiation (a perfect greenhouse).
