Physics of the Greenhouse Effect Pt 1

Jean Baptise-Fourier is generally credited with the discovery of a greenhouse effect, which involves the process by which the presence of an atmosphere acts to raise the surface temperature of a planet. This was extremely simplified at the time, and the term greenhouse never appears in his 1827 writing, but he did establish the effect that the atmosphere had on incoming light and outgoing infrared (heat) radiation, and that some heat was absorbed by the atmosphere which was opaque in the infrared but transparent to incoming solar energy. A copy of his essay, translated by R.T. Pierrehumbert can be found here. We’ve made a lot of progress since then, as Svante Arrhenius began to quantify the phenomenon nearly 75 years later, the work of Stefan and Boltzmann established the relationship between an object’s temperature and its outgoing radiation, the role of convection and water vapor and clouds turn out to be important in more complex models developed later, etc. The pionerring paper by Arrhenius, entitled On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground, which is the famous 1896 piece, began to investigate what the effects of doubling the atmospheric CO2 content would be. At this time, most of the interest in the subject was in solving the mystery of the coming and going of ice ages. Like most pioneering efforts, Fourier or Arrhenius did not have the last word, and we still have much to learn today, but they provided a big leap in how we understand planetary temperatures and the role of the atmosphere in radiative balance. Fourier was one of the first to speculate that human activities could influence climate, and such topics are rather important in modern times.

Notwithstanding the ancients who thought the Earth sits on the back of giant animals, it is well known that the Earth sits in the large vacuum of space. Because there is no physical medium by which the Earth can receive sunlight, it must travel some 149,600,000 km in the form of electromagnetic radiation. Because the planet is also not in physical contact with anything, the only way in which it can lose heat is by radiation. For an outer skin temperature of 5770 K, the sun’s energy output is 6.3 x 106 W/m2. Spread over the area of the Earth’s disk facing the sun, the Earth receives roughy 1,368 W/m2 (S) of solar radiation. Since the planet radiates like a sphere, to average this value over the whole Earth, the energy received by the Earth’s disk (πr2) needs to be divided by the surface area of the Earth’s sphere (4πr2). This results in ~342 W/m2 of solar energy at the top of the atmosphere. Roughly 70% of that is available to heat the planet, as the remaining is reflected right back to space (technically speaking, the planet has an albedo of 30%, or 0.3). Overall, the Earth and atmosphere receive ~239 W/m2 of solar energy to keep us a warm and drive ocean and atmosphere circulation.

All objects above absolute zero emit radiation, and the intensity of that radiation depends on the objects temperature. More specifically, that intensity rises proportional to the fourth power of the objects temperature (σT4 where σ is Stefan Boltzmann constant, at 5.67 x 10-8 W/m2 K4, and T is in Kelvin). The Earth is no exception, and in order to lose heat to space, it must balance the incoming solar radiation with its own infrared radiation at the top of the atmosphere. If it did not emit radiation back out to space, it would only gain heat and eventually heat up without bound. Any object that is not in radiative equilibrium will warm or cool depending on if its received more energy than it emits, or emits more than it absorbs, but the temperature change is how the object comes back to balance.

Recall that the electromagnetic spectrum is divided into several parts, such as in the infrared, visible, UV, microwave, etc. In this discussion, we will deal only with visible (also shortwave radiation) and infrared (also called longwave radiation). Visible radiation is more intense than infrared radiation, and is emitted by objects at higher temperatures. So far, all we need to get is that the sun preferentially emits in the visible, while the Earth loses heat preferentially in the infrared.


Just to get a bit of this out of the way, the effective temperature of the Earth with radius r is:

πr2(1 – a)S= 4πr2σT4, or

Teff = [S(1 -a)/4σ]1/4 = 255 K

We can then write an equation for the energy balance of the atmosphere, as

Iup,atmosphere + Idown, atmosphere = Iup, ground

= 2ɛTatmosphere4 = ɛTground4 (also accounting for emissivity), or

Tground = fourthroot (2Tatmosphere)

This temperature is below freezing, and so this shows that if the Earth’s temperature were purely based on the amount of solar radiation it receives, it would be far from habitable. The gap between our present day comfort, and an iceball planet is due to the fact that some of the outgoing infrared radiation is not immediately sent right back to space, but is absorbed by the atmosphere, where some is radiated downward to the surface. This is due to the fact that we have greenhouse gases, which are transparent to incoming solar radiation, but absorb outgoing infrared radiation strongly. The mean surface temperature difference is,

Δ T ≡ Ts – Teff = 33 K

The mean temperature of the Earth’s surface is actually 288 K, which says that the greenhouse gases are responsible for a 33 K enhancement. No longer freezing, but rather comfortable and unique to the solar system.

So what is going with this greenhouse??

The following image shows a spectra at the top of the atmosphere which shows the absorption of photons by CO2, water vapor, ozone, etc.


a more detailed image can be found at

There are several greenhouse gases in the atmosphere including carbon dioxide, water vapor, methane, ozone, nitrous oxide, and some which are purely human-made (anthropogenic) such as CFC’s. These gases absorb the longwave (IR) radiation emitted by the relatively warmer surface and emit radiation to space at the colder atmospheric temperatures, leading to a net trapping of IR energy within the atmosphere (the greenhouse effect). The long-term climate of our planet is governed by a balance between the incoming solar radiation, the reflected solar radiation (albedo changes based on land cover changes, presence or absence of ice sheets, etc) and the emitted IR. If one increases the amount of infrared absorbing gases, then excess energy is suddenly available to drive the climate system.

Now there is a popular misconception that more greenhouse gases warm the surface on the pure basis that they enhance the downward infrared emission. I want to examine what is going on in a bit more detail, because saying that the globe is warming simply because the downward IR has increased is not accurate. The below figure shows how a quick profile of the atmosphere.


As we can see from the Y axis, pressure decreases with height, and so does temperature. The temperature decrease with altitude should be familiar to anyone who has ever climbed a mountain, and is known as the lapse rate. The temperature gradient between the surface and the cold atmosphere gives rise to to atmospheric motions that contribute to vertical transport of heat, such that the air temperature decreases about 6.5 °C per kilometer. If the pressure of the surrounding air is reduced, then the rising air parcel will expand. If the total amount of heat in a parcel of air is held constant (no heat is added or released), then when the parcel expands, its temperature drops because of the inverse relationship between the volume of an air parcel and its temperature. Put simply, because of gas expansion principles the air temperature decreases with height. If this were the whole story, one would expect the decrease to be nearly 10°C per km (the air parcel is moving along the dry adiabat). The mean lapse rate is less than the dry adiabatic value because of latent heat release by condensation as moist air rises and cools and because the atmospheric motions that transport heat vertically include large-scale atmospheric dynamics as well as local convection. For more on the lapse rate, see Tamino’s blog post here. The surface receives much more solar radiation than the atmosphere (recall that the atmosphere is mainly transparent to incoming shortwave radiation), and so convection arises to put excess heat into the colder surface; as it turns out, the troposphere can be thought of as more or less being constrained by convection to stay near the moist adiabat and warms and cools more or less as a unit. This fact, along with the idea of the lapse rate will turn out to be the key behind the greenhouse effect.

Going back to the above diagram, let Prad denote the pressure at which the effective radiating level is (i.e. where radiation escapes to space). The outgoing longwave radiation (OLR) is equal to the net incoming solar radiation in order for the planet to be in radiative equilibrium. It is very important to think of the planet’s radiative balance as being defined at the top of the atmosphere, rather than at the surface. The radiative (and non-radiative processes like convection) are shown in this diagram


To satisfy the second law of thermodynamics, the *net* energy flow is from the surface to the atmosphere. Now let’s just move past a simple example of “greenhouse gases warm because of more downward energy.” On net, the radiation from the surface is greater than the back-radiation from the greenhouse effect. Greenhouse gases moreso slow the *net* energy flow from the surface to the atmosphere, and the surface and atmosphere to space. For example, if 10x units of solar radiation hit the surface, then

10x solar in, 10x infrared out establishes radiative equilibrium

Now suppose that the solar is 10x to the surface, but only 6x of infrared escapes to space, and 4x is absorbed by the atmosphere. Of that 4x, 2x is re-radiated back to the surface, and 2x radiated back out to space.

So now

12x in the surface
8x back out to space

This is a case of where the surface is not in radiative equilibrium, and what is happening is that the surface cooling is inhibited . Ultimately, the surface is warmer than would otherwise be in a no atmosphere case because it is heated by both the sun and the atmosphere. For such an imbalance to be rectified, the temperature must increase until as much infrared goes out as solar is received. Let’s go back to the figure of our atmosphere (the one with pressure, and temperature decrease).

Now, let’s throw a greenhouse gas like CO2 into the atmosphere between the surface and Prad. This means that the layers below Prad will be optically thick and strongly opaque in the infrared. Because CO2 absorbs at certain wavelengths, there will be a bite taken out of certain places in the atmospheric window where outgoing radiation would otherwise have a free trip back out to space. For CO2, it just so happens that the wavenumber 667 cm-1 is strongly absorbed, and so radiation at this wavelength will be absorbed in the atmosphere. Adding more greenhouse gases renders the atmosphere more opaque to the outgoing infrared radiation, and so it is not so easy to get out. Radiation will travel layer-by-layer in the atmosphere, and will interact with various layers in various types of ways. Some may collide with neighboring air molecules to warm that layer such that all the air molecules in a neighboring region are in thermodynamic equilibrium, it may be radiated downward, or it might go upward. Since the temperature decreases with altitude and since colder bodies radiate less efficient than warmer bodies, some of the upwelling radiation wil finally get to an area so cold and thin that it escapes to space. In the graph, this altitude is roughly 5 km above the Earth’s surface.

An estimate of the greenhouse warming is

Ts ~ Te + Γ H

where H is the flux-weighted mean altitude of the emission to space and Γ is the mean lapse rate between between the surface and the level H. Now suppose that we decide to come along and throw a few more CO2 molecules into the air. This means that the lower levels become more opaque to the outgoing radiation, making it harder to escape. This forces Prad to move up in altitude (i.e. go to higher layers), or you decrease Prad.

This is where the argument that an enhanced greenhouse effect causes warming because the enhanced downward emission becomes incomplete. Remember that it is the top of atmosphere energy budget that is affected when we alter the planet’s radiative budget. The surface energy budget drags along as a secondary role, and determines the gradient between the surface and atmosphere. In fact, it may turn out that enhancing the downward radiation does little if evaporation and sensible heat fluxes respond accordingly to not make it much a difference. When you have a very moist surface, it is very difficult to get a large temperature gradient between the surface and overlying air. In another example, suppose that the lower atmosphere was so opaque to infrared radiation that it radiated like a blackbody at its temperature. In this case, adding more greenhouse gases will not make it anymore of an emitter because it is already a perfect emitter. Consider the two cases in the below picture.

In the left case, we have an enhanced downward pertubation by adding more greenhouse gases, but also a surface with plenty of room to evaporate more. Turbulent fluxes of moisture and heat also exchange energy between the surface
and the atmosphere, and these become dominant especially when the radiative term is weak, or may enhance to respond to the enhanced IR perturbation to close the surface budget. More evaporation will close the surface energy budget, such as to make enhancing another term (like the new downward IR) relatively insignificant. In the right case, adding more CO2 will not cause warming because of “enhanced IR” because there is no more room for making the blackbody slab anymore opaque to infrared. Any CO2 above the blackbody layer will be absorb radiation before it gets back to the surface. This may be the case in an area with heavy cloud cover or saturated with water vapor. Because of various responses by the surface energy budget, it may be that if some researcher was looking into the effects of how increasing the downward emission was affecting surface temperatures in the tropics, they may come to the conclusion that the enhanced greenhouse effect was not a significant factor. This may lead to the idea that the enhanced greenhouse effect is not longer able to rise temperatures, or to the idea that CO2 is already saturated, but both conclusions would be wrong.

In the present times, the planet is now receiving more energy than it emits back out to space. Such a planetary energy imbalance has been predicted and confirmed (e.g., Hansen et al., 2005). In addition, the oceans are now taking in heat which is where most of the heat from the imbalance goes. Because of the large thermal intertia of the oceans, there is still a measurable amount of warming that is “in the pipeline” which means that even if modern CO2 concentrations were held constant, we would warm up a bit more. The direction of this planetary energy imbalance mentioned is opposite the imbalance that we would expect if modern day global warming was caused by internal variability such as ocean circulation changes, etc. But it would be miselading to say that the heat content change is due to re-radiated IR. The global mean heat content change is the result of changes in *net* radiation reaching the ocean surface, which includes downward re-radiated IR due to CO2, water vapor, clouds, etc., but also includes any increase in the *absorbed* solar radiation at the ocean surface due to any long-term decrease in clouds. But I should be more clear on what happens when you change the greenhouse composition of the atmosphere.

By making the lower levels more opaque to infrared radiation, you decrease Prad and so you are forced to extrapolate temperature in the vertical further along the adiabat to reach the surface. This is because the effective radiating level has now shifted to higher levels where the planet can emit the radiation back to space. Recall that temperature decreases with altitude, and radiation diminishes with temperature. This means that making the lower levels more opaque in the infrared, you decrease the rate by which the Earth can get rid of heat by forcing the the mean altitude from which infrared emitted upwards makes it to space to ever higher altitudes. These are colder layers, and so radiate more feebly. The net inlux of solar radiation remains constant, while the planet is now radiating at colder temperatures, and so less efficiently. Taking from the chart in my post just a few more molecules based on MODTRAN, we can see how the OLR diminishes accordingly.


So now, the planet is taking in solar radiation at the same rate as before, but losing heat less efficiently. This means the *net* influx of radiation is positive, and so the planet is taking in more solar radiation than emitting back to space. Since the troposphere warms as a unit, the lower layers must warm up until this heating imbalance disappears. It no longer matters if we put in a hypothetical “blackbody slab” in the lower level of the atmosphere, because we are not increasing the emission by making it more opaque to infrared, but by increasing its temperature. In this way, we see that global warming must happen until the top-of-atmosphere energy budget is restored to equilibrium.

We can see that the greenhouse effect means that the planet radiates at a colder temperature than the surface. So it is important to realize that the greenhouse effect depends on the fact that the temperature decreases with altitude. No lapse rate = no greenhouse effect. If we introduce more greenhouse gases, then there must be some cold air aloft for the greenhouse gas to work with. This is certainly the case in the troposphere.

As one last emphasis, the greenhouse effect is not the same thing as global warming. If we are specifically talking about human-induced global warming, than that refers to the enhanced greenhouse effect (along with other factors like deforestation and other land use changes). The greenhouse effect itself is naturally occurring, and is a necessary condition to keep the surface temperatures warm enough for life. Global Warming specifically refers to the *change in* conditions from pre-industrial time to today.

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13 responses to “Physics of the Greenhouse Effect Pt 1

  1. Excellent explanation Chris. I also feel saying that the net radiation down being more than zero, rather than purely the IR down is a better way of putting it. A matter of the intended audience I suppose.

  2. To the author of this post,

    I am a high school Earth Sciences teacher, and we will be discussing some of this topic in my class soon. I may use some of this material as a foundation, as it was instructive. One of my students who enjoys this topic and reads your site referenced me here, and I also learned some things. Keep up the good work.

  3. Chris, I had to re-read this a few times, and I have a few questions. if the upper atmosphere warms as a unit then why is the upper troposphere now cooling? Second, you said in another post that we’re getting 1 W/m2 of an imbalance (the Hansen paper is about the same), but the IPCC says greenhouse gases are something like 2.4 W/m2 forcing now. What is wrong?

    Response– Nope, the stratosphere is cooling, not the upper troposphere. I will have to do a post on that soon. The number you gave from the IPCC is the GHG forcing alone, the total *net* forcing is less because of the cooling effect of aerosols. The net forcing is ~1.6 W/m2. The rest of the gap is because that is an equilibrium number, not the instantaneous number. If everything remains constant and you let the oceans respond and the system to come back to equilibrium (which takes decades), then you have the IPCC number.– chris

  4. Pingback: Hypography Science Forums - My belief in Global Warming is getting shaky

  5. A very interesting post. Thanks.

    A couple of nitpicking points:
    1. In paragraph 3, I think the units in your Boltzmann constant needs a / i.e. W/m^2T^4.
    2. In the formula just below the Electromagnetic Spectrum graphic, should T be T^4? And I assume r=R.

    Response– Thanks, the units are fine, but I fixed your point 2.– c

    • The units are ambiguous – that is what he meant, I suspect. A much better way is to forget division bars and use just positive and negative indices viz, Wm^-2 K^-4

    • The units are ambiguous is what I suspect he meant to say. Rather than use a ‘ / ‘ it is less prone to mis-interpretation if you use negative indices. viz. Wm^-2K^-4

  6. Chris:the IPCC puts RF at 1.66. So to get the real amount of forcing I must subtract the negative values from Ozone and aerosols which would make1.65
    as the net amount. The net warming of the sun seems minute @.12. Is this right.At Accuweather we have a lot of arguments about GCM’s and their ability to predict. From every part of the world, there seems to be an agreement that in 2100 the temps should rise from 2.6C to 5C. Logic presumes that that if all the models that are used from every scientific group
    in the World that deals with predictions is right than barring any major external event, this should be true. Do you agree? Yes, and I would like to use your global warming piece as the place to go for understanding this topic.Dennis Hlinka told me about you as I talked to him about a month ago. Thanks,KIPP

    Response– The *greenhouse gas* forcing is 2.5 W/m**2 or something around there (From memory) and the negative forcings from aerosols push it back to 1.6 for the net effect. All of the forcings are relative to 1750 so a positive RF for the sun may not mean it’s been positive for the last few decades (it’s actually been slightly negative), but it was going up earlier in the 20th century.

    Models need to be treated with a good amount of skepticism, and for some things they probably aren’t very reliable such as regional precipitation trends. In many cases (like Arctic sea ice melt) models have substantially underestimated the extent loss. In general, you’re right, the more different models from different groups agree the more robust the conclusion is. The global temperature response to given changes in external forcings is pretty good, but cloud feedbacks are responsible for much of the 2 to 4.5 C range you speak of…that’s a fairly big range but there’s a lot of evidence that the range is correct, and if you don’t like models, the paleoclimate record very clearly shows that the climate is not insensitive. The same people who claim it is are the same ones that shout about how much it changed in the past, or how negligible changes in the sun can have such a big influence.– chris

  7. There are three main issues with “global warming” that are never mentioned in the press but which will become quite obvious over the next couple of years. This is most certainly NOT the time for rash action.

    Most of the warming of the 80′s and 90′s can be directly attributed to the Pacific Decadal Oscillation. A persistent, multidecadal pheonomenon that can be shown through the temperature record. In warming mode it leads to frequent El Ninos. In cooling mode it leads to frequent La Ninas. From about 1880 to 1910 it cooled, then it warmed until about 1940, cooled until about 1980 warmed until about 2002. The PDO has recently switched back. Factoring in the PDO it looks unlikely that the total increase by 2100 would reach 1C even if CO2 were a factor. Feel free to look up on wikipedia or other sources.

    Urbanization is another important factor, but not in the way one might think. There are four main “groups” dealing with global temperatures. The one most frequently used is GISS (thank you Dr. Hansen) which uses surface station data. Over time urbanization around these surface stations has lead to increasing bias. Many have gone from green pastures to asphalt parking lots. Some are even beside HVAC heat exchangers. It is likely that a great deal of the remaining warming is due to these issues. There also happen to be two satellites. At the time of their launch the satellites showed temperatures .2C lower. They’re now approaching .4C lower temperatures than GISS. The urbanization probably began after WWII but the affect was hidden by the cooling phase. An independent surface station audit is now being done (you should check it out, the state of our surface station system is just…tragic)

    The last important factor is the sun. The sun is the source of all of our heat in the first place and it’s activity seems to have been the reason for the warming after the little ice age. Indeed, graphing global temperatures v/s solar cycle length (a good indicator of the strength of the solar dynamo) the match is uncanny…and goes back hundreds (technically thousands) of years (not the pitiful 20 year period of “correlation” for CO2 that ended about 10 years ago).

    While total output of the sun normally only varies by .1% over the highs and lows of a typical solar cycle the spectrum varies quite a bit. During the unprecedented activity of the 20th century the sun’s UV output varied by as much as 4-8%. Every bit of that extra energy (a significant part of the sun’s spectrum) was absorbed and turned into heat by the ozone layer. Now the sun has slumped into a level of inactivity unseen in the modern age. The solar wind and magnetic fields have dropped by 30% or more. The sun’s light output has also dropped more than usual..probably most of it coming from that UV part of the spectrum.

    There’s also a suggestion that cosmic rays let in by the sun’s now weakened magnetic field may seed cloud formation, resulting in additional loss of heat from increased albedo. The science is anything but settled on “global warming”. Indeed, 2008 saw the largest temperature drop in recorded history. We should find out over the next year or two just how powerful the affects of the sun and PDO really are and also how (in)accurate our surface station data has been.

    Response– You must be reading some very lousy sources. There has been no trend in solar activity or cosmic rays for over half a century now, and the PDO (by definition) cannot contribute to the long-term trend.

  8. Thanks for an extremely readable and clear explanation of the Greenhouse effect.

    Now I have a problem of understanding which I hope you can explain. I understand that the energy radiation balance is determined at the top of the atmosphere, and the surface temperature is determined by the lapse rate causing a temperature gradient between the surface and the top of the atmosphere.
    Now comes the question. If the surface temperature rises, this should also increase the amount of water vapour. However, we know that increasing the water vapour content decreases the lapse rate, meaning that if the temperature of the top of the atmosphere is fixed, a reduced temperature gradient means that the surface temperature will drop. Doesn’t this amount to a negative feedback?

    Response– Well remember that the height level of constant temperature is shifted to higher altitudes, so I’m not really sure that is a good argument. That said, it does turn out that you get a negative lapse rate feedback because of the moisture increases higher up, and the fact that the tropical atmosphere stays roughly close to a moist adiabat. This does act to partially offset the water vapor feedback. This is discussed in post here. It’s a relatively small effect, but it does matter (see the lapse rate is negative in the last diagram of that post).– chris

  9. That matches what I suspected – that negative feedback from lapse rate is offset by the height of constant temperature rising in altitude in response to the increasing GHG effect, and in addition there is also the positive effect of the water vapour feedback. That adds three plusses and minuses into the equations. It sounds it should be possible to express this in differential equations. I’ll have to take the time to study the link you gave. I’m interested to see that this is an extremely hot topic right now with Lindzen’s latest bombshell.
    Thanks for the fast reply, congratulations on an excellent site, I like the balance of readability and solid technical content!


  10. You are quoted as saying, “Response- Nope, the stratosphere is cooling, not the upper troposphere. I will have to do a post on that soon. The number you gave from the IPCC is the GHG forcing alone, the total *net* forcing is less because of the cooling effect of aerosols.” In response to Lloyd’s comment on a reduced surface temperature from the decrease in lapse rate and temperature gradient, you said, “Response– Well remember that the height level of constant temperature is shifted to higher altitudes, so I’m not really sure that is a good argument.”

    What higher altitudes are you referencing? The stratosphere begins a warming trend until it transitions into the next layer of the atmosphere. Assuming constant temperature rises to a particular altitude is a misnomer for the earth because the layers thin as you approach the poles from the equator. Can you explain why temperature would remain constant while moving up in the atmosphere?

  11. “Since the planet radiates like a sphere, to average this value over the whole Earth, the energy received by the Earth’s disk (πr2) needs to be divided by the surface area of the Earth’s sphere (4πr2). This results in ~342 W/m2 of solar energy at the top of the atmosphere”

    There is the flaw in the whole “science” of the “greenhouse” hypothesis. The average temperature of earth without greenhouse gases would be minus 18 C is nonsense – the average is meaningless and reduces this argument into the absurd.

    The earth’s actual blackbody temperature would be much higher during the day and probably much much lower at night – the artificial reduction of the insolation to 342 W/sq metre is voodoo science – with this absurdly low figure how do you explain temperatures on earth of in excess of 50 + C – back radiation ? Give me a break – the sooner we start teaching real meaningful science the better.

    The fact that this whole sham is built on an artificial reduction of solar insolation from 1368 W/sq metre to 342 W/sq metre is ludicrous.

    Water with its ability to exist in three phases at ambient temperatures found in various places on earth and the energy required to cause the phase changes is the driving force behind climate. The effects of CO2 are at most almost negligible compared to the role water plays.

    Response– So all points of the planet, at any given time, are receiving the same amount of sunlight? — chris

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