# Physics of Global Warming

1. Feb 22, 2009

### Xnn

The temperature of the earth is governed by physics, namely the Stefan-Boltzmann law which states that the amount of energy radiated is proportional to the fourth power of its temperature.

Where:
SB, the Stefan-Boltzmann constant is 5.670 x 10-8 Watt/meter^2 Kelvin^2

For Earth at equilibrium, the amount of energy radiated should equal the amount of energy received from the sun. However, the Earth is not at equilibrium and is actually receiving slightly more energy that it is emitting. This is why the earth is warming. If the earth were in equilibrium, then the amount of energy being radiated would equal the amount received from the sun. That is ERad would be a constant and a function of average Total Solar Irradiance (TSI) and albedo (a).

TSI is 1365.5 Watts/meter^2
a, albedo which is 0.3 for Earth

So, ERad is approximately 237 Watts/meter^2. Putting this altogether yields an Earth Temperature of 254°K (-18°C or -1°F). This temperature corresponds to the atmospheres temperature at about 5 kilometers above the surface (16,000ft). It is at this elevation where the earth radiates to outer space approximately the same amount of energy it receives from the sun.

Temperatures at lower elevations are generally much warmer due to the greenhouse effect, which makes it difficult for the atmosphere to radiate infrared energy at lower elevations. In fact, greenhouse gases inhibit radiation to such an extent, that convection of heat is the dominate mechanism for transporting energy from the surface to elevations where it can be effectively radiated to outer space. Keep in mind that the earth radiates primarly in the infrared which is the predominate wavelength at 254°K. Infrared is invisible to humans.

Anyhow, if there were no greenhouse gases, then earths surface temperature would become so cold that the oceans would freeze. This in turn would raise the earths albedo and reflect more energy directly to outer space. In turn the Stefan-Boltzmann law would drive the temperature even colder and we would end up living on a giant snowball.

However, the earths atmosphere does have greenhouse gases. In particular CO2, which warms the atmosphere enough so that water can exist as a vapor. Since water vapor is also a greenhouse gas, together these greenhouse gases have warmed earths surface to about 287°K (14°C or 57°F). While CO2 may comprise just a small fraction of the atmosphere, it behaves like a dye in that it absorbs infrared energy very well.

Finally, the earths temperature is not in equilibrium. The earth is absorbing about 1.5 watt/meter^2 more energy than it is emitting. This in turn is warming the atmosphere, oceans, land, snow and ice. By far, most of the extra heat is going into the oceans. The oceans have a tremendous capacity for storing heat and it will take a long time before they reach equilibrium. When equilibrium is eventually reached, there will be more evaporation of water and the atmosphere will become thicker from increased amount of water vapor. This will result in warmer surface temperatures and a higher elevation at which the earth can radiate to outer space.

I’m not the first person to figure this all out. In fact, an intergovernmental panel of climate change scientist (IPCC) have been studying this subject intently for well over 20 years. The IPCC has carefully reviewed many scientific studies and have published their latest assessment here:

http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-ts.pdf

What has been concluded (TS.4.5 on page 64) is that the earths temperature is sensitive to changes of CO2 concentration. In particular, equilibrium change is likely to be in the range 2°C to 4.5°C per doubling of CO2, with a best estimate value of about 3°C.

Last edited: Feb 22, 2009
2. Feb 22, 2009

### Gokul43201

Staff Emeritus
Several points of dissatisfaction already, at this juncture:

1. What is the emissivity of the Earth? Why doesn't it figure in the first equation? Or is it assumed to be equal to 1?

2. It is distracting to use \epsilon for albedo, when that is the symbol typically used for emissivity.

3. Where does the factor of 1/4 come from? Your derivation makes no mention of it, and only someone that had done the derivation independently or seen it done elsewhere will understand its origin. Presumably, it comes from actually doing the flux calculation (using Gauss' Law or taking dot products and integrating - either way will give a factor of 4 between the effective radiating surface area and the effective absorbing surface are, barring albedo).

4. Clearly, the temperature extracted here is some kind of average. What kind of average it is is obscured by the extent of hand-waving involved in the calculation. For instance, since half the planet receives virtually zero insolation at any point of time, so you are integrating over a period of some multiple of a one day to arrive at this number. Then, the insolation varies by latitude, even for the day half of the planet. But it looks like the integration over latitudes (not explicitly done) precedes the radiation calculation (which implicitly assumes that temperatures equilibrate across latitudes instantaneously, doesn't it?), rather than the other way round.

Personally, I find the above derivation to be too hand-wavy and irrigorous to be helpful. I have to read between lines and fill in too many missing steps, if I want to understand anything from this.

Last edited: Feb 22, 2009
3. Feb 22, 2009

### Xnn

1. Emissivity averages about 0.62 for the earth and can be used to calculate surface temperature. It basically takes into account the greenhouse gas effect. As CO2 levels rise, emissivity will lower and surface temperature go up.

2. Sorry; I have edited it to an "a" in order to avoid confusion.

3. 1/4 is just a geometry measure that comes from the ratio of the area of a globes disk to it's surface area: pi r^2 to 4 pi r^2. pi r^2 cancels out leaving just the number 4. The earth receives radiation from the sun in proportion to the cross sectional area of its disk. However, it radiates energy over the larger area of it's surface. Dividing by 4 takes into account the geometry of a sphere.

4. No hand waving; just simplification. A completely accurate earth climate model requires lots of programming and a super computer to run. What is presented is just the simple overall physics for global warming. The temperature is what would be expected using the Stefan-Boltzmann law for Earth. 33°C (the delta between 287 to 254K) is the amount of warming on the surface from the greenhouse gases.

Last edited: Feb 22, 2009
4. Feb 22, 2009

### WeatherRusty

Excellent opening post!

"The earth is absorbing about 1.5 watt/meter^2 more energy than it is emitting."

One quibble. Where does the 1.5W/m^2 figure come from. Hansen is famous for claiming 0.85W/m^2

5. Feb 22, 2009

### Xnn

Thanks WeatherRusty!

The 1.5 watt/meter^2 comes from 2 places.

First, satellite mission called the Earth Radiation Budget Experiment and the Earth’s Radiant Energy System (CERES) as well as sensors on NASA’s Terra and Aqua satellites have directly measured the imbalance. They have found it to be about 1.4 watts/meter^2. Of course it varies over time.

Second, the IPCC in the Technical Summary on page 32 Figure TS.5 has calculated it and reported a value of 1.6 watts/meter^2 with a range of anywhere from 0.6 to 2.4 watts/meter^2.

So, theory and measurements are in good agreement.

Last edited: Feb 22, 2009
6. Feb 22, 2009

### John Creighto

I'm not convinced at this part:

Why is Є the albeto. What I'd say Є is, is is the fraction of the outgoing radiation that gets back radiated. You could call it an albedo but it is the albedo looking back out to space for IR radiation at 5KM altitude.

With regards to the greenhouse effect I think the altitude of interest would be between 9km and 17km because bellow that convective forces control the temperate gradient.
http://www-das.uwyo.edu/~geerts/cwx/notes/chap01/tropo.html

The problem is that radiation emitted back into space is not emitted at a single point, It is radiated from various altitudes at various frequencies. Also if you want to talk about the radiation emitted from the atmosphere remember that the atmosphere is not a black body because it is not optically dense. It consequently has an emissivity less then one and therefore you have to multiply the right hand of Stephan Boltzmann equation by the emissivity.

7. Feb 23, 2009

### Tanja

I have a comment and a question on the solar constant TSI:

The magnetic field and the solar wind of the sun are changing in a 11yrs cicle and also on a monthly and even daily basis. This magnetic field must be coupled in a certain way to the sun's temperature, since the magnetic fields arise from plasma movements which should be driven be temperature and pressure gradients. Therefore the Planck's function of the sun can actually not be constant due to the changing temperature. And the integral over the Planck's function, the so called "Solar Constant" can't be a constant.
Is there something wrong with my conclusion or is the name "constant" just missleading? Anyway, if the solar constant is used as a constant in climate models, I'm not surprised that they are wrong.

8. Feb 23, 2009

### Phrak

Prove it.

9. Feb 23, 2009

### Xnn

John Creighto;

Correct; radiation is emitted at various locations and frequencies. Emissivity is basically a measure of this and is effectively the greenhouse effect. What has been presented is ignoring the greenhouse effect in order to calculate the differance between reality and what surface temperature would be without it.

Tanja;

1365.5 Watts/meter^2 is a typical value for Total Solar Irradiance. It does indeed change over time by about 1 watt/meter^2. Application of the Stefan-Boltzmann equation shows that this amounts to about 0.05 C.

Phrak;

First; a satellite mission called the Earth Radiation Budget Experiment and the Earth’s Radiant Energy System (CERES) as well as sensors on NASA’s Terra and Aqua satellites have directly measured the imbalance. They have found it to be about 1.4 watts/meter^2. Of course it varies somewhat over time.

Second; Temperature measurements of the atmosphere and oceans as well as melting sea ice, and the Greenland and Antarctic ice caps have shown that all major parts of the earth are warming.

Third, the IPCC in the Technical Summary on page 32 Figure TS.5 has calculated the inbalance and reported a value of 1.6 watts/meter^2 with a range of anywhere from 0.6 to 2.4 watts/meter^2.

10. Feb 23, 2009

### WeatherRusty

The Solar constant or Total Solar Irradiance (TSI) is not a constant at all. Over the 11 years sunspot cycle it is measured by satellite to vary by 0.1% or 1.3W/m^2. This is enough to cause a 0.1C (+-0.05C) change in temperature at Earth's surface. Since the mid 1700's proxies indicate the Solar irradiance to have increased by 0.1% to 0.2% overall. It has not increased since satellite readings have been taken, beginning in 1979 I believe.

When the Sun is more magnetically active the proportion of ultraviolet radiation reaching Earth is strongly enhanced, but most of this is absorbed in the upper atmosphere and thus not available to warm the surface directly.

http://www.physorg.com/news129483836.html" [Broken]

http://en.wikipedia.org/wiki/Solar_variation" [Broken]

Last edited by a moderator: May 4, 2017
11. Feb 23, 2009

### Xnn

By the Stefan-Boltzmann equation, 1.3 watts/meter^2 is 0.06C. No +-

12. Feb 23, 2009

### John Creighto

The emissivity is not the measure of the greenhouse effect. It is a measure of optical density. Low optical density implies and emissivity near zero. Large optical density implies an emissivity near one. The greenhouse effect is a result of the optical density being different for IR radiation then visible radiation. For an object to be a black body it must have an emissivity near one.

13. Feb 23, 2009

### WeatherRusty

The Moon is at the same average distance from the Sun as Earth and so receives the same 1366W/m^2. It has an albedo of .12 as opposed to Earth's .30............but harbors no meaningful atmosphere.

Mean surface temperature (day) 107°C
Mean surface temperature (night) -153°C

Lunar Mean surface temperature: -23C

Earth's radiative equilibrium temperature: -18C (15C-33C greenhouse effect)

Why might Earth without an atmosphere be 5C warmer than the Moon without an atmosphere even though the Moon is the darker (albedo) object? The oceans? The oceans are the great reservoir of accumulated heat which maintain a higher than equilibrium temperature near Earth's surface. Where does the additional accumulated heat come from? The greenhouse effect! The warmed atmosphere baths the oceans in thermal radiation warmer than an atmosphere containing no greenhouse gases. Night time temperatures drop very little over the liquid oceans because of the large heat carrying capacity of the water, maintaining a higher average diurnal temperature than over a solid surface.

14. Feb 23, 2009

### Xnn

John;

Emissivity is the measure of an object's ability to emit radiation energy compared to a black body at the same temperature. In the case of a atmosphere, it is largely a function of the concentration of gases that can absorb and emit energy in the infrared. These gases are commonly called greenhouse gases...

15. Feb 23, 2009

### Skyhunter

The solar constant is not constant. In fact in all the universe the only constant is flux.

However, since average TSI flux over time is ~0, a constant value for TSI is used in simpler models.

The models are not right or wrong, they are tools for understanding complex chaotic systems.

16. Feb 23, 2009

### WeatherRusty

The atmosphere emits because it has a temperature. The solar photosphere emits because it has a temperature. Both are composed of gases radiating thermal radiation produced during the collisions between the atoms and molecules in bulk matter. The Sun (mostly hydrogen & helium) predominantly in the visible because it is at 5,780K, the Earth's atmosphere in the infrared because it is at 255K. You can think of it as frictional heat, some of the kinetic energy of the gas atoms and molecules is radiated away when the atomic and molecular structure is disturbed during collisions with others. This means that oxygen and nitrogen are radiating IR, though by a different mechanism than the greenhouse gases do at specific wavelengths.

17. Feb 23, 2009

### Xnn

The moon rotates very slowly which also contributes to the extreme differances between day and night. To calculate its temperature using the Stefan-Boltzmann equation, I'd be tempted to not use the 1/4 geometry factor and just figure the max temperature. In doing so, I come up with a value of 108°C, which is reasonably close to the mean day temperature quoted. So, I suspect that those temperatures are closer to the max and mins as opposed to an average or mean value.

Unfortunately, there aren't a lot of good references to check and since the day/night temps are so extreme, it may not be an easy task to measure the average temp of the moons surface.

18. Feb 24, 2009

### Phrak

These data sources are science when practitioners are in a position to be objective. As the last is governed by a political body, it fails to meet this screening.

Provide critical sources for any of your remaining claims. Not the original reports, but the criticisms--pro or con if you can.

Last edited: Feb 24, 2009
19. Feb 24, 2009

### Skyhunter

The IPCC is a valid source. It does not conduct science, it assembles the body of research related to climate change every five years into a comprehensive assessment.