Mark concluded that my comment: “appears to boil down to this key statement: “more CO2 reduces emissivity from the surface from (say) 0.615 to (say) 0.605”.

My comment said that, IF YOU USE A GRAYBODY MODEL, a doubling of CO2 is equivalent to a reduction of emissivity from 0.615 to 0.605. I also said that a graybody model is a flawed model for our climate system, because Planck’s Law and the S-B equation only apply when absorption and emission are in equilibrium. They aren’t at many wavelengths and altitudes in our atmosphere. “Effective emissivity” is merely a fudge factor that allows our planet to emit only 240 W/m2 given a surface temperature of 288 K. In reality, none of the components of our atmosphere has an emissivity this low and the temperature of emitting molecules ranges from 190 to 310 K.

Despite these problems, a graybody model is commonly used and gets some things approximately right. It is also full of bobby traps that can confuse you. I tried to show you how a graybody model can be transformed into something that looks like conventional climate science with “feedbacks” that come from recognizing that da/dTs (the change in albedo with the change in surface temperature) and de/dTs (the change in effective emissivity with surface temperature) are not zero. If rising CO2 is equivalent to a decrease in emissivity, then absolute humidity that rises with Ts makes de/dTs non-zero (along with other LWR feedbacks). When others misuse graybody models, you hopefully have a chance of spotting their problem.

The danger of useful analogies like Brad’s or graybody models is that they all fail at some point if you push them too far. Other analogies – such as saying GHGs act as an insulator, because some people don’t understand that insulation blocks heat transfer by both convection and radiation. My favorite climate cartoon has a picture of a thermometer next to a block of ice and asks if the presence of ice makes it warmer. Yes, a block of ice makes it warmer if empty space a 3 K lies on the other side of the ice!

My take home message is that Schwarschild’s equation contains the physics/chemistry that correctly describes the interaction between radiation and GHGs as it travels through our atmosphere. Everything else is a crude approximation. The terms n, o, T, I, and dI vary with altitude and technically should be written as functions of altitude n(z), o(z), T(z), I(z) and dI(z) (and ds becomes dz using the two stream approximation). Since it is a differential equation that has no analytical solution and must be integrated over all wavelengths to get W/m2, it can be difficult to use if you didn’t understand my intuitive interpretation. If your intuition doesn’t encompass this mathematics or if you need quantitative answers, you need to rely on software that automatic numerical integration of this equation: Modtran, Spectracalc, or the results found in papers that do more sophisticated calculations. And this blog has many excellent posts in the “Understanding Atmospheric Radiation and the Greenhouse Effect” and the “Visualizing Atmospheric Radiation” series that illustrate the surprising phenomena Schwarzschild’s Equation generates. For example:

Final note: Radiation has an intensity, but doesn’t have a “temperature”. However, radiation intensity can be converted into a “blackbody-equivalent temperature” using the S-B equation. When people talk about the “temperature of radiation”, they may be referring to this concept. They may also mistakenly use that temperature in connections with the 2LoT. Individual photons don’t obey the 2LoT. Instead the net two-way flux between any two locations always obeys the 2LoT, because large numbers of photons and molecules are following the laws of quantum mechanics. (A single molecule has kinetic energy that is constantly changing due to collisions, but not a “temperature”, a collective property of a large group of colliding molecules.)

]]>Frank, thanks again for your explanation yesterday 19/8 at 10.27 pm.

It appears to boil down to this key statement: “more CO2 reduces emissivity from the surface from (say) 0.615 to (say) 0.605”.

This is a bit like Brad’s analogy of shining a torch into a fog (or indeed the more obvious example of being under low lying cloud at night).

Fair enough, that’s either true or it isn’t, and if true, are there other mechanisms that automatically cancel it out (radiation is not thermal energy any more that potential energy or latent heat is thermal energy. Radiation of course converts into thermal energy, in which case it’s not radiation energy any more, is it?).

I shall have to think about that.

]]>Mark: I hope you recognize the Schwarzschild equation as more than “interesting” – as an accurate representation of the physics/chemistry that determines how radiation interacts with GHGs in our atmosphere. This physics is what the online Modtran radiation transfer calculator linked by DeWitt uses, our host used in many posts to illustrate the non-intuitive behavior of radiation in simple model atmospheres, and the IPCC relies upon calculate radiative forcing with today’s atmosphere. (AOGCM’s also rely upon these calculations, but they also rely on tunable parameters that are not the product of rigorous science.) Furthermore, the quantitative predictions of the Schwarzschild equation and our database of absorption cross-sections measure in the lab have been validated by numerous experiments in our atmosphere. Most of all, I hope that your dilemma about the doubling of emission associated with doubled CO2 (which I shared) disappeared and you understand what really happens to both absorption and emission when GHGs increase (before anything else changes).

You can make some rough approximations about what the temperature of planets using:

absorbed SWR = outgoing LWR

(S/4)*(1-a) = eoT^4

I’ll let you and DeWitt debate those calculations. (I always like including the sunlit side of non-rotating Mercury, which is much cooler than Venus, in any list of planets I claim to understand.) Remember however, that Planck’s Law and the S-B equation technically shouldn’t be applied to our atmosphere because absorption and emission aren’t in equilibrium at many wavelengths and altitudes. Choosing the correct “effective emissivity” (e) for Earth is a little tricky. If you use surface temperature (288 K), e = 0.61 and if you more accurately assume e = 1 for the atmosphere, you get a temperature of 255 K, about the temperature of the atmosphere where the average photon escaping to space is emitted. The difference between these two temperatures is caused by the lapse rate and a 5 km difference in altitude. The lapse rate in the troposphere is determined by convection, not radiation, which is why you don’t need radiative transfer calculations to explain this temperature difference. These black- and gray-body models do give different values for Planck feedback.

However, you need to recognize that when 2XCO2 slows the rate at which our planet radiatively cools to space (as shown by Schwarzschild’s equation), that is equivalent to reducing the effective emissivity of our planet from 0.615 to 0.605 in a graybody model (a 1.5% change). Or equivalent to raising the altitude from which the average photon escaping to space is emitted by 0.170 km in a blackbody model. And you need to remember that “small” but persistent changes in radiation build up over time to amount to very large amounts of energy. A +1 W/m2 radiative imbalance at the TOA is enough to warm the ocean to a depth of 50 m by 0.2 K/yr. (50 m is the depth warmed and cooled by seasonal changes in irradiation.) And a 1 W/m2 forcing from increased CO2 can persist for decades. I doubt any crude methodology you use for comparing planetary temperatures will be relevant to GHG mediated climate change.

Mark wrote: You don’t need to worry about whether thermal energy is transmitted by conduction, convection and radiation. It is safe to assume that energy in all its forms does its best to even out everywhere.

This is incorrect. When heat is transferred vertically only by convection, you get a stable linear lapse rate that falls altitude at a rate of Cp/g (which needs to be corrected for the heat released by condensing convected water vapor). In a gray model of an atmosphere, when heat is transferred only by radiation, you get a stable linear lapse rate that changes with optical density, which means it rises exponentially (nonlinearly) approaching the surface. That lapse rate is unstable toward bouyancy-driven convection until the tropopause. And conduction alone produces an isothermal atmosphere. So the mechanism of vertical heat transfer does matter. Since convection transfers heat vertically in the troposphere (outside of polar regions) faster than any other process, we observe a linear lapse rate there, but not higher. We also see no fractionation of gases by molecular weight in the troposphere, showing that bulk motion is mixing the troposphere.

Today we have an imbalance at the TOA because heat retained by rising GHGs hasn’t stopped flowing into the ocean. If you insist on applying crude models to our planet, I suggest starting by restating the above equation in terms of a radiative imbalance (I) at the TOA and surface temperature (Ts) and then differentiating:

I = (S/4)*(1-a) – eoTs^4

dI/dTs = -(S/4)*(da/dTs) – 4eoTs^3 – (oTs^4)*(de/dTs)

You’ve got a term for Planck feedback, SWR feedbacks as albedo changes with Ts (as we discussed above), and LWR feedbacks as rising humidity decreases the effective emissivity of our planet by acting as a GHG and by changing the lapse rate in the atmosphere so that the upper troposphere warms faster than the surface. And clouds change with changes in Ts, effecting both albedo and emitted LWR. Of course, this is just the standard formulation used by climate scientists expressed in terms of a graybody model for the planet. Not being physicists, they have given fancy new and misleading names (feedbacks and amplification) to simple derivatives involving albedo and emissivity. (:))

]]>Mark,

So why is it so easy to reconcile surface temp on Mars, Earth, Moon and Venus without this tweak?

That’s completely bogus. Ignoring radiative transfer in a planetary atmosphere does not explain planetary surface temperature. It has been discussed at length elsewhere on this site.

]]>Frank, that’s all very interesting.

So why is it so easy to reconcile surface temp on Mars, Earth, Moon and Venus without this tweak?

All you need to know is

– distance from Sun

– height of cloud cover and albedo

– mass and height of atmosphere

– specific heat capacity of gases in atmosphere

– strength of gravity

– rotation speed (affects the Moon in particular, as it has no atmosphere to even things out).

You don’t need to worry about whether thermal energy is transmitted by conduction, convection and radiation. It is safe to assume that energy in all its forms does its best to even out everywhere.

(For Earth you have to adjust for latent heat of evaporation as well)

]]>Mark: You asked a great question: How does a flux of energy (power/unit area) become a temperature which is proportional to energy? It doesn’t! An energy flux in W/m2 becomes a RATE of temperature change. You also need to know the depth of the material being heated through its surface area, which give you the volume being heated. Heat capacity (J/volume/K) then allows you to convert a W/m2 flux into a rate of warming K/s. Or you need to know the mass of the material being warmed through its surface are and use J/kg/K for heat capacity. Volume works for warming the ocean. Mass works for warming the atmosphere.

In the case of our planet, we talk about the radiative imbalance created at the TOA by an instantaneous doubling of CO2 (a radiative forcing). As the planet warms in response to reduced radiative cooling to space, OLR will increase and the radiative imbalance will gradually shrink to zero. If you integrate that radiative imbalance (power) over the time, you get energy – which is found in the internal energy of our warmer climate system

]]>Mark wrote: “I just fail to see how INCREASING the amount of gas which can radiate IR to space can REDUCE the amount of IR radiated to space. That’s like arguing if your car has a bigger radiator, the engine will get warmer.”

I struggle with this problem for a long time. Twice as much CO2 will emit twice as many photons and those photons will travel on the average half as far between absorption and emission. There will be no change in the flux to a first approximation. However, the traveling half as far argument doesn’t apply to photons that would have escaped to space before doubling. I didn’t understand radiative forcing until I learned the proper physics.

The emission and absorption of radiation (photons) is the controlled by the laws of quantum mechanics, which simplify to “Schwarzschild’s equation for radiation transfer” in the stratosphere and troposphere (in the absence of scattering, which is usually negligible for thermal IR, but not visible):

dI = emission – absorption (+/- scattering)

dI = n*o*B(lambda,T)*ds – n*o*I*ds = n*o*[B(lambda,T)-I]*ds

where dI is the change in intensity of radiation at a wavelength lambda (spectral intensity) as it passes an incremental distance ds through any material, I is the spectral intensity of the radiation entering the increment ds, n is the density of molecules absorbing and emitting radiation (GHGs in the case of thermal IR), o is their absorption cross-section at wavelength lambda, I is the spectral intensity of the radiation entering the increment ds, T is the local temperature, and B(lambda,T) is Planck’s function. This equation is used to calculate the forcing from 2XCO2 by integrating the thermal IR emitted by the surface as it travels from the surface to space. DLR is calculated traveling from space to the surface. Individual photons don’t obey the Second Law of Thermodynamics, but the net flux of radiation calculated using Schwarzschild’s equation is always from warmer to colder.

The first time SOD showed me this equation, the “lights went on” and all my confusion about radiative forcing disappeared. I don’t know if the implications of this equation are immediately obvious to others. SOD has written many posts showing the non-intuitive results that are produced by numerically integrating this equation.

When radiation has passed far enough through a homogeneous isothermal atmosphere, eventually absorption will come into equilibrium with emission. dI will be zero and I = B(lambda,T). Planck’s law can be considered to be a corollary of Schwarzschild’s equation and Planck derived his law explicitly assuming radiation where absorption and emission are in equilibrium with “quantized oscillators”. Planck’s law doesn’t apply in the atmosphere, because absorption and emission are not in equilibrium at all wavelengths and altitudes. This is why our planet’s OLR spectrum doesn’t look like that of a blackbody.

If really high intensity radiation enters the ds increment, the emission term is negligible and Schwarzchild’s equation simplifies to the differential form of Beer’s Law. A laboratory spectrometer uses a filament at several thousand degK to produce light so intense that thermal emission is negligible. Using laboratory spectrophotometers causes many people to ignore emission and think only in terms of absorption. Both absorption and emission comparable in the atmosphere.

The implication of Schwarzschild’s equation can be simplified to: The spectral intensity of radiation traveling through an atmosphere is being changed by absorption and emission so it approaches “blackbody intensity” at a RATE proportional to the density of absorbing/emitting molecules and the absorption cross-section (n*o). If you take the most strongly absorbing wavelength for CO2, 90% of the photons emitted upward from the surface (with near blackbody intensity) have been absorbed in the first 1 m – and they have been replaced with an equal number of photons emitted upward by CO2 molecules. If you put a low emissivity coating on the surface so that it emitted no radiation up, upward emission by from CO2 molecules will have produced a blackbody intensity within a few meter of the surface. And if you shone a laser tune to this wavelength upward from the surface, absorption would dominate emission and the upward flux would also have blackbody intensity a few meters above the surface. By the time radiation gets well into the stratosphere, the density of CO2 molecules has been reduced to the point where radiation can travel a long distance between absorption and emission and the temperature can change over that path. If temperature changes, temperature-independent absorption and temperature-dependent emission may no longer be in equilibrium.

Now let’s consider a horizontal layer of atmosphere ds thick in the middle of the troposphere. Photons are emitted from this layer in all directions, but we can break that emission into three parts: a component in the +z direction, a component in the -z direction and a component in horizontal directions that contributes nothing to heating or cooling the planet and can be ignored. (This is called the two-stream approximation.) Emission from our layer of atmosphere contributes equally to dI in the +z direction and the -z direction. Now, what about absorption. Radiation entering this layer is also traveling in all directions and again we only need to consider the components traveling in the +z and -z directions. Since radiation traveling upward was emitted lower in the atmosphere where it is usually warmer, upward I is greater than the local B(lambda,T) emission. Therefore, the [B(lambda,T)-I] term is negative and dI is negative. When you add up all of these dI terms from the surface to the edge of space, the 390 W/m2 upward flux at the surface is reduced to 240 W/m2 (OLR). More importantly, when the concentration of GHG (n) is increased, dI gets more negative, explaining why rising GHGs reduce radiative cooling to space. For radiation traveling in the opposite direction, dI is usually positive and the downward radiative flux increases from zero at the edge of space to about 333 W/m2 at the surface (DLR).

To be completely candid, increasing GHGs (n) also reduces the magnitude of the [B(lambda,T)-I] term, because photons travel a shorter distance when the concentration of GHGs are higher and the temperature difference between the altitude where incoming radiation (I) was emitted and the layer is emitting B(lambda,T) is smaller. For this reason, the wavelengths where photons travel a long distance between emission and absorption make the greatest contribution to radiative forcing.

And if you are really sharp, you will have noticed that there would be no GHE or enhanced GHE (radiative forcing) with an isothermal atmosphere whose temperature didn’t change with altitude. And why rising CO2 cools the stratosphere.

]]>I just fail to see how INCREASING the amount of gas which can radiate IR to space can REDUCE the amount of IR radiated to space. That’s like arguing if your car has a bigger radiator, the engine will get warmer.

That’s because you don’t understand radiative transfer. Emission originates at the surface, but only a small part of that radiation escapes directly to space. Ghg’s emit and absorb radiation. That atmospheric emission is proportional to the temperature and concentration of the gas. Temperature and concentration (pressure) decrease with altitude. At 400 ppmv and wavelengths near the center of the CO2 band, effectively all radiation emitted from below is absorbed at an altitude of a few meters. That continues as altitude increases until the number density of CO2 molecules (number of molecules/m³ or pressure times length: atm cm e.g.) gets low enough that 50% of the radiation emitted upward can escape to space. That’s the effective emission altitude. The same sort of thing applies to the band wings. The result is that, because temperature in the troposphere decreases with altitude, the intensity of the emitted radiation decreases as the concentration increases because the effective altitude of emission increases.

If you want to see how that works, try going here:

http://climatemodels.uchicago.edu/modtran/

You can vary all sorts of parameters and see what happens.

If you set all the variables except viewing altitude to zero, you get a blackbody emission curve with a power of 380.254 W/m² at a surface temperature of 288.2 K. The emissivity is not one (0.98 IIRC) and some energy at long wavelengths is not included in the integration so it’s less than σT^4. Now let’s add some CO2 to the atmosphere:

10 ppmv 364.554 W/m²

20 ppmv 360.786 W/m²

50 ppmv 356.076 W/m²

100 ppmv 352.622 W/m²

200 ppmv 348.854 W/m²

400 ppmv 345.4 W/m²

Emission decreases non linearly with increasing CO2. The same goes for water, methane, tropospheric ozone and freon.

Note that initially the CO2 absorption band gets wider and deeper, but at about 50 ppmv, the effective emission altitude for the center of the band is in the stratosphere where the temperature increases with altitude so you get a peak in the middle of the dip.

Grant Petty’s textbook isn’t expensive ( in fact, it’s currently on sale for the original price of $36) and can give you a good grounding in the fundamentals of meteorology and radiative transfer.

]]>… If you have MORE emitters… Sorry, auto correct.

]]>Mark, another thought in trying to drive this point home – if you have now emitters in a system you also have more absorbers. This is why increasing concentrations doesn’t simply equate to more emissions, as you’ve also increased the number of absorbing particles above the original emitting layer of the atmosphere.

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