Every once in a while it is worth reviewing the basic physics behind the greenhouse effect and global warming. Sometimes all the debate about global warming in the media loses focus of the fact that the world really is governed by the laws of physics. Unfortunately, many internet explanations get dumbed down to the point of having an atmosphere that serves as a single “slab” between the ground and space, and has a bunch of colorful arrows coming out of it and bouncing off it, etc. This is a useless explanation, and gives no justice to understanding what is happening. Two encounters in the outside world recently prompted me to do another post just to have a reference handy, and I’m using this to replace an older post which I entitled “just a few more molecules.” There’s also been an interesting episode with Dr. Andy Lacis from NASA GISS over at Dot Earth which I’d like to elaborate on.
We begin with the Planck function, which describes the radiation emitted from a blackbody at a specified wavelength and temperature:
This has physical dimensions of intensity (power per unit area per unit solid angle) per unit wavelength, often in W m-2 µm-1 steradian-1 (a steradian is essentially the 3-D analog of what angles are in two-space; there are 4π steradians in a sphere). h, c and k are constants, λ is the wavelength, and T is the temperature. An important note is that dB/dT > 0 for all wavelengths, which suggests that increasing the temperature increases the intensity at all wavelengths.
For review, the electromagnetic spectrum is presented below
For our purposes, we’re mostly interested in the fact that the Earth receives energy in the shorter wavelength portions of the spectrum, while it loses its energy in the infrared portions of the spectrum. This distinction between incoming and outgoing energy is made easy for planets such as Earth, where there is essentially no overlap and the two can be treated independently. This would not be the case on very hot planets that could radiate at some several thousand degrees K.
It can be difficult to interpret the Planck law simply by looking at the formula, so it is worth plotting the Planck function to visualize.
As we’ve established, the first thing to note is that the total area under the curve increases as temperature increases, corresponding to increased emission at each wavelength and in total as the object becomes hotter. We can also see that the peak wavelength shifts to shorter values as temperature increases. The wavelength of maximum emission can be obtained by setting and solving for the wavelength, which gives Wien’s law (~ 2897 µm K / T). The Planck law thus helps us understand why the Earth emits primarily in the longwave, infrared portion of the spectrum, while the sun emits primarily in shorter wavelengths, much of which is in the visible and near-infrared portion of the spectrum.
We can integrate the Planck function over all wavelengths to get the total power output per unit area, which gives us the Stefan-Boltzmann law. We also include a factor of to account for the solid angle integration over a hemisphere.
where is the Stefan-Boltzmann constant, approx. 5.670 x 10-8 W m-2K-4.
We’re now in a position to formulate the most basic radiative balance equation for planets. We can assume the no-atmosphere Earth radiates like a blackbody, although non-perfect radiators emit like where represents the ratio of how well the emitter is to a perfect blackbody. At equilibrium, this radiative balance is:
where is the ratio of the surface area of the planet to the cross-section (which is very close to 4 for planets), (1- ) is the fraction of absorbed solar radiation (since is the albedo=reflectivity) that comes from the solar constant S0, and R is the distance to the star in astronomical units. It automatically takes on a value of one for the Earth-sun mean distance and S0 takes on a value of 1370 W m-2. This radiative balance equation states that, at equilibrium, the Earth wants to lose as much infrared radiation to space as it absorbs by the sun. This basic calculation assumes no atmosphere and a uniform planetary temperature. This uniform temperature assumption is nearly valid for planets such as Earth or Venus, but is way off the mark for those which exhibit extremely strong diurnal gradients like Mercury or the moon.
Present Earth albedo is 0.3, so the calculated equilibrium temperature for our planet would be 255 K. This is well below the freezing point of 273.15 K. This planet would be frozen down to the tropics! Since the sun is the only important incoming source of energy to Earth, internal heating plays a negligible role in the planetary energy balance (although is important for gaseous planets like Jupiter), then we can say that the difference between the observed 288 K and 255 calculated temperature is due to the fact that the atmosphere acts to inhibit the efficiency by which outgoing infrared radiation escapes to space. In radiative terms, this is:
This post is not meant to discuss why greenhouse gases are greenhouse gases. But it should be noted that the bulk of the atmosphere, including N2, O2, and Argon are not infrared active molecules. It is actually very common, although perhaps not well known, for diatomic molecules to become good greenhouse gases in pressure-induced planetary situations (this is important on Jupiter for instance, and on the dense atmosphere of Titan), but aside from their role in broadening the lines of the other greenhouse gas molecules, they play essentially no role in Earth’s radiation balance. The absorbing gases discussed in this post include water vapor, clouds (which aren’t gases, but still contribute to the greenhouse effect), CO2, methane, N2O, ozone, and a variety of other lesser important constituents.
The rest of this post is basically a description of the manner in which absorption by these molecules affect the planetary energy budget. Before that happens, a quick review of the temperature structure of the atmosphere. It can be shown quite easily that the temperature of the atmosphere drops with height. For air devoid of water vapor, this rate is . For a moist atmosphere, this rate is reduced to about -6.5 K km-1 due to latent heat release of condensation, which results in the temperature still decreasing, but less rapidly than that rate given by the dry adiabat. The real atmosphere at a given time is generally somewhere between a dry and moist adiabat.
We now have the tools to look at some model emission outputs. As I’ve done before, I’ll use the MODTRAN model available from Prof. David Archer’s web page.
The settings I will use in all experiments is to keep the sensor at 70 km looking down, U.S. 1972 Standard Atmosphere, and for the sake of instruction, no clouds, water vapor, or rain. Let’s begin by constructing a plot in which CO2, CH4, and ozone are all set to zero. We obtain:
What the plot shows is a set of colored line which correspond to blackbody curves at specified temperatures. Note that we’re now looking at wavenumber on the x-axis, which is the inverse of wavelength, so the peak on the graph now shifts to the right for increasing temperature. 10,000 divided by microns will give you wavenumber in inverse centimeters, which are common units used in this case.
The red squiggly line corresponds to the Earth curve with the specified settings. The model is fixed at 288.20 K unless changed in the input settings, so the red curve is somewhere between the 280 and 300 K blackbody curves, though not emitting perfectly like a blackbody. We see that Iout is 346.97 W m-2. Now let’s add just 2 parts per million of CO2 into the atmosphere.
It is useful in atmospheric radiation to describe an optical thickness (or optical depth), along a vertical path,
where k is an absorption coefficient, and is the density of absorber. The transmittance through an absorbing medium goes like .
The noticeable difference in the second figure is that there is now a “bite” taken out of the Outgoing Longwave Radiation (OLR) curve. Viewed from space, an observer wearing infrared goggles would now see emission emanating from colder layers of the atmosphere. In parts of the spectrum where the atmosphere is optically thick, the radiation to space occurs at the temperature of the high, cold parts of the atmosphere. On the other hand, the transmitted ground emission will dominate the OLR when the atmosphere is fairly transparent.
In the 2 ppm case, the OLR is now 338.8 W m-2. Since the area under the curve has been reduced due to the blip caused by CO2, we are now interested in re-establishing the outgoing flux to be 346.97 W m-2. We can do this by raising the temperature in the model by 1.9 C. Simply through fundamental physics, it can be shown that the temperature must increase as greenhouse gases are added to the atmosphere. Now let’s re-set the ground T offset to zero and put in 50 ppm of CO2 into the model. The resulting graph is:
If the greenhouse gas in question were absorbing only in this limited interval, then increasing its concentration further could not bring down the OLR any further, since in the spectral region where the gas is radiatively active, the atmosphere is already radiating at the coldest temperature possible. Technically, since the stratosphere cools with more greenhouse gases, this would have a minor effect on the OLR, but it is a negligible one for our purposes. If we change the amount of CO2, the intensity of light in this range does not get any lower. This is called band saturation.
Instead, further increasing CO2 will decrease the OLR essentially by filling the “wings” of the spectral bands where the atmosphere is optically thin. We can see this by putting in a modern concentration of 390 ppm CO2. The manner in which the “wings” of spectral lines become important is discussed in Dr. Ray Pierrehumbert’s RealClimate post on Angstrom.
Greenhouse gases have absorption whose overall strength decays rapidly with distance in wavenumber from the central peak, which helps explain the reason in which the OLR change goes like the logarithm of CO2 amount at concentrations relevant to the global warming problem. Here is a plot of the outgoing energy term as a function of CO2 concentration. This is a visual confirmation that after you have already introduced some CO2 into the atmosphere and are up to realistic concentrations, every doubling will produce the same radiative forcing.
One could compute the temperature change for a perturbation in the OLR for just a change in CO2, leaving other climate variables constant.
which gives a ~1 C rise in temperature for a 4 W m-2 radiative forcing. This would be the temperature response to a CO2 doubling if only the Planck radiative feedback were important. Unfortunately, life is not that easy, and you need to figure in the feedbacks from water vapor, clouds, albedo, etc.
Now let’s put some crazy amounts of CO2 into the model, say 10,000 ppm.
As we can see, the bite in the spectrum is not getting any deeper, but it is getting considerably wider. This is very important for the greenhouse effect, since you will essentially never become “saturated” and thus you will keep getting warming with more and more CO2. We should now be in a position to interpret a plot of atmosphere transmission or absorption,
This helps to visualize the percent absorption by atmospheric constituents on a scale from 0 to 1 as a function of wavelength. It is important to note that the absorption coefficient is a strong function of wavelength. Although radiative transfer equations would often be much easier to solve if the optical thickness was independent on wavelength (the so-called grey gas approximation), alas, this is not how life operates. This is somewhat convenient though, since if the atmosphere behaved like a grey gas, changing CO2 from a “nice concentration” just a tiny bit would make the planet virtually inhabitable, setting an extremely narrow window for proper atmospheric conditions suitable for life.
Let’s go back to our model outputs obtained from MODTRAN. Note that there is a small “blip” at the bottom of the center CO2 band, which becomes very apparent in the 10,000 ppm case. This is a signature of the stratosphere, where temperature increases with height. You get this blip in a very high CO2 case since the atmosphere is so absorbing in the center of the band that emission is coming from above the tropopause, while in the wings, where absorption is weaker, emission comes from below. This “blip” is potentially useful to planetary scientists looking for life on distant planets, since a hot stratosphere could be a signature for ozone, and thus oxygen. This blip would probably not be real if you were to really add that much CO2, since the stratosphere cools with more CO2 as it becomes a less effective absorber but a better emitter of radiation. You can somewhat see this if you lower the sensor altitude to 30 km or so.
It is really important (in fact, essential) to understand that the greenhouse effect requires colder air aloft to work with, as you essentially replace strong surface radiation with weaker emission from higher layers. This is why adding CO2 creates an energy imbalance at the top-of-atmosphere. In an isothermal atmosphere, you could not get a greenhouse effect.
The effective height “H” above is obviously dependent on wavelength, and moves higher as the opacity in increased in that region.
So…review: Because of energy balance, the planet must get rid to space as much energy as it receives from the sun. Averaged over the Earth, taking into account the albedo and geometry, this is about 240 W m-2. In the absence of an atmosphere, this flux of radiation is lost by the surface by . With an atmosphere, this flux of radiation is allowed to emanate from upper, colder layers of the atmosphere, say on average at some altitude H. Increasing greenhouse gases increases the altitude of H, a height in the atmosphere which depends on wavelength, and characterizes a level of mean emission to space. Because the atmosphere is now emitting from colder levels of the atmosphere, the OLR has decreased, and the result is that the planet must warm to re-establish radiative equilibrium.
Radiative transfer in the earth’s atmosphere is not particularly amenable to simple formulas because the atmosphere is semi-transparent to differing degree at different wavelengths. For example, there is a popular formula for a 2-layer equilibrium relationship which is obtained by treating the atmosphere as a slab and solving multiple equations for the energy balance of each slab. This states that Ts = 21/4Teff (or generalizing, Ts = (n+1)1/4Teff, where n is the number of layers). It can be found, for example, in David Archer’s “Understanding the Forecast”, or Dennis Hartmann’s “Global Physical Climatology.” As an exercise, I urge readers unfamiliar with this model to try to derive this expression using principles discussed here. You can also refer to these lecture notes (slide 16, 17 on radiative-convective equilibrium) if you need help. This is valid for an isothermal grey absorbing layer above a Planck emitting ground, and applies only when the layers are either totally transparent or totally opaque. Such a model can be made more complex by allowing for non-unity absorptivity/emissivities in different layers, allowing for shortwave absorption in the atmosphere, etc. These lend very good insight into the nature of temperature inversions, geo-engineering, how one might need to later the albedo to offset CO2 increase, etc. One might find out for instance that if we perfectly offset enhanced absorptivity by increased reflectivity to keep the surface temperature constant, you can still get a change in atmospheric temperatures, which would thus have implications for the lapse rate and stability. So, I urge experiments with these approaches. Although this sets the stage for basic textbook explanations to get a feel for radiative balance, in GCM’s, the radiative transfer problem needs to be addressed numerically, with a sufficient number of vertical layers to resolve the atmospheric temperature and absorber distribution, with an acceptable amount of spectral intervals to resolve the spectral dependence of the contributing gases, and account overlap between various atmospheric constituents. Further, using a 2-layer model in a fully transparent shortwave and fully opaque longwave atmosphere achieves a surface temperature of 303 K, which introduces another obvious problem with this model, notably the lack of convection which removes heat from the surface, and also establishes the vertical temperature profile of the atmosphere from which the greenhouse effect relies on.
In modern concentrations, every doubling of CO2 will reduce the OLR by about 4 W m-2. This doesn’t hold at very low concentrations as we’ve seen, but also at very high concentrations. The phenomena of band saturation also allows us to say something interesting when comparing two greenhouse gases side-by-side.
It’s often noted that methane is “20x more powerful than CO2” (see a quick google result for proof). This statement can potentially be misleading, so it is worth clarifying just what it means.
The natural 33 K greenhouse effect has a much larger influence from CO2 than it does CH4. Even in the context of how the greenhouse effect is changing, CO2 is currently a much stronger forcing agent than CH4. In what sense is CH4 more powerful? This is only if we compare CO2 and CH4 side-by-side and allow the two gases to change by some incremental amount at existing background concentrations. It is only because CH4 is far less abundant in the atmosphere that adding, say, 1 ppm of CH4 will produce a larger radiative forcing than would be adding 1 ppm of CO2 to today’s atmosphere. This has nothing to do with any intrinsic property of the gas. If CO2 were far less abundant, and CH4 much more abundant, then adding a certain about of CO2 would be more effective at reducing the OLR, and we would then say “CO2 is a more powerful greenhouse gas.”
“The most important greenhouse gas”
Dr. Andy Lacis of NASA GISS is a radiative transfer expert and has worked on planetary climate stuff for a long time now. He has recently been hailed by global warming skeptics for a criticism of an early draft of the IPCC summary for policy makers. Lacis had plenty of words to say to the “skeptics” about this at Dot Earth (linked at the top of the post, you can follow the several posts about it) but finally Lacis felt the need to simply go back and review the basic physics of radiative balance which constrains the global climate. Along the way, Lacis is quoted as saying
“The bottom line is that CO2 is absolutely, positively, and without question, the single most important greenhouse gas in the atmosphere. It acts very much like a control knob that determines the overall strength of the Earth’s greenhouse effect.”
This generated some interesting comments, but let’s examine what Dr. Lacis was getting at, since I certainly agree with this.
If you break down the natural greenhouse effect into a fractional contribution between constituents you end up getting that about 50% of the 33 K greenhouse effect is due to water vapor, about 25% to clouds, 20% to CO2, and the remaining 5% to the other non-condensable greenhouse gases such as ozone, methane, nitrous oxide, and so forth. It is therefore popularly noted, from popular discussion to textbooks to academic papers that “water vapor is the most important greenhouse gas” in the atmosphere. While it is true that water vapor represents the largest source of infrared opacity in the atmosphere, the claim that it is “most important” is just sloppy terminology with no real attempt to define what exactly that is supposed to mean. At some level, this could just be academic disagreement, but it is worth putting into perspective that CO2 is indeed the control knob which governs the global climate.
The non-condensable greenhouse gases (e.g., CO2, CH4, etc) are those which do not precipitate from the atmosphere at Earth-like conditions. They can support a temperature of nearly 10 K above that which is determined by equilibrium with the incoming solar radiation on their own. Water vapor, on the other hand, is controlled by temperature. I have a rather long post discussing the radiative feedbacks which are important for understanding the sensitivity of climate, which defines the temperature response per unit radiative forcing. The amount of water vapor in the atmosphere is not set by sources and sinks, but has an upper limit on its concentration until it precipitates out. Because of this, the non-condensable greenhouse gases act as the backbone upon which water vapor and clouds can really do their stuff. Without them, you would also get a near collapse of the terrestrial greenhouse effect as water vapor content declined nearly exponentially with a declining temperature. This scenario would cause a runaway snowball Earth and a much higher surface albedo. Indeed, removing either water vapor or CO2 from the atmosphere would trigger glaciation down to the tropics. When thinking about the evolution of Earth’s climate though, it is CO2 which changes, and water vapor which is dragged along to amplify the total response.