Do Additional Greenhouse Gases Warm or Cool the Earth?

If the Earth’s atmosphere had no infrared-active gases, commonly and confusingly called greenhouse gases, at all, the Earth would be colder on average.  The Earth’s surface would absorb more of the sun’s insolation, since water vapor would not be present to absorb the incoming energy from the sun and there would be no clouds.  Some of the heat absorbed by the surface would still be transferred to the nitrogen, oxygen, and argon molecules or atoms striking the Earth’s surface.  The remaining energy would be radiated from the surface.  Virtually all the energy radiated from the Earth’s surface would travel at the speed of light through the atmosphere into space and be almost instantaneously lost.  The day to night temperature changes would be much more dramatic than they are now.  The greenhouse gases benefit us greatly by moderating the day to night temperature changes.  At sufficiently low concentrations, each infrared gas with a non-overlapping absorption frequency with respect to other infrared-active gases already present, will slow down the rate of cooling at the surface and in the troposphere.  This allows the surface and the troposphere to be warmer than they would be were the infrared-active gas not present.

This is how the idea of a greenhouse effect comes about.  This paper Is not disputing that infrared active gases allow the Earth’s surface to be warmer than it would be if they were not in the atmosphere.  The question being examined is whether the further addition of an infrared-active gas will warm or cool the Earth’s surface and its lower atmosphere, the troposphere, when its atmospheric concentration is increased.

How does a low concentration of an infrared-active gas significantly slow down the cooling rate of the surface and the troposphere?  Suppose this gas molecule absorbs the longwave thermal radiation emitted from the Earth’s surface in the lower troposphere and enters an excited vibrational state.  If that absorbed energy were simply immediately re-emitted and carried off the previously absorbed energy at the speed of light, the absorption event would have no significant effect on the temperature of the surface.  The key fact here is that the excited molecule has billions of collisions per second with the 2500 times as plentiful non-infrared-active molecules of nitrogen and oxygen and atoms of argon.  That absorbed infrared energy is converted into kinetic energy passed to the molecules that collided with the excited molecule long before the lifetime of the excited molecule for re-emission of the absorbed energy by radiation.  What is the overwhelmingly dominant means of energy transport through the troposphere at this point?  It is the convection transport from warmer to cooler portions of the atmosphere, which is generally upward and to the higher latitude regions of the Earth.  The speed of that transport of energy is about 8 orders of magnitude slower than the speed of light.  So that first act of longwave thermal radiation absorption from the Earth’s surface is of immense importance, but after that initial conversion of infrared radiation from the surface into kinetic energy shared by all the molecules (mostly nitrogen and oxygen) and atoms (argon mostly) of our troposphere, the role of any further radiation from infrared-active molecules is a faster means of cooling than is the convection cooling mechanism.

Let us examine why this is true.  If the mean free path for absorption of a given wavelength of the longwave thermal radiation from the surface is short enough that there is an absorption event by an infrared molecule in the lower troposphere, subsequent absorptions of any emitted thermal radiation by molecules at that wavelength at higher altitudes will prevent that radiation from escaping into space.  This does slow down the cooling of the lower atmosphere in this very limited context.  This is what the standard view of greenhouse gases focusses on.  The problem is that the alternative to that infrared-active gas emitting thermal radiation to a higher altitude is its being much, much more slowly transported to higher altitude by convection.  Adding more of that infrared-active gas to the atmosphere results in moving energy upward through the troposphere in steps at the speed of light instead of having more of it moving upward in the slow convection currents.  The implication here is that a very low concentration of an infrared gas in the atmosphere will produce a warmer Earth surface and troposphere, but subsequent additions simply speed up the transport of energy to higher altitudes.  Then the added infrared emitting gas molecules in the upper troposphere and the stratosphere radiate thermal energy at a faster rate directly into space.  For an infrared-active molecule, the initial effect of adding it to the atmosphere is likely to be warming effect, but its warming effect rapidly passes through a maximum and then further additions start bringing down the temperature at the surface and in the troposphere.

Let us pause and get a better understanding of how carbon dioxide absorbs longwave radiation from the surface of the Earth.  An infrared-active molecule has an absorption spectrum over a range of wavelengths with absorption probabilities varying with wavelength over orders of magnitude.  The absorption probability at a wavelength is usually given in terms of an effective cross section, as though the size of the molecule were different for the absorption event at each wavelength.  Here, from a figure in Prof. Howard “Cork” Hayden’s The Energy Advocate, February 2020 (Vol.24, No.7) is the absorption spectrum near the main 15 µm (micrometer) absorption line for CO₂:

As Prof. Hayden explains, if the concentration of carbon dioxide in the atmosphere were only 40 ppm by volume (ppmv) or a bit less than one-tenth the present concentration, any radiation at the wavelengths above the red line in the figure at 1 x 10^-22 m² cross-section would be absorbed within a travel distance of 10 meters.  At that same very low concentration of CO₂, the absorption distance for radiation in the weaker absorption peaks above the lowest red line is less than 100 meters.  The troposphere in the U.S. Standard Atmosphere is 11,000 meters in altitude.  All parts of the absorption spectrum above the lower red line for as low a concentration of carbon dioxide as 40 ppmv are already absorbed many times traveling through the troposphere from the surface to the upper troposphere.  At 400 ppmv, carbon dioxide will add absorption events for the first time in the lower cross-section parts of the spectrum, but the additional first time absorptions are decreasing rapidly as more CO₂ is added to the atmosphere, while the rapid transport effects of carbon dioxide are moving more and more energy to space more quickly than convection would at an increasing rate as CO₂ is added.

I am going to develop a simple model for how the infrared-active gases warm the Earth, according to those who believe in catastrophic or even moderate man-made global warming.  This is not a model that I believe is correct.  This is intended as an exercise in determining a very generous upper limit on the size of the greenhouse gas warming claims based on additions of such gases to the atmosphere above those of the present and then showing that those claims are false in the context I have set up above.

I will start with a two layer atmosphere in which radiation from the Earth’s surface, is absorbed entirely in the lower layer L1, which radiates half that energy upward to atmospheric layer L2 where it is absorbed and half back down to the surface where all of it is absorbed.  All of the surface absorbed half from L1 is re-emitted upward and is re-absorbed in L1.  Layer L2 emits half the energy it absorbed from L1 directly to space, where it is permanently lost, and half back to L1 where all of it is absorbed.  Each layer L1 and L2 always emits half of its absorbed radiation energy upward and half downward.  The surface always emits its absorbed energy back to L1, where it is absorbed.  I believe this isotropic emission idea is incorrect, but we are going to do this exercise because it will give us insight and because it is rather a fun task to work out.  Note also that I am entirely ignoring convection as an energy transport mechanism for the sake of this argument.

The results are that space receives a series of emission energy originating with an emission of one unit of energy from the Earth’s surface, Sp, whose sum is

So, all of the unit of energy emitted from the surface is eventually emitted into space.  In fact, the first 5 emissions to space already total 0.7627 of the total energy of 1.  In this crazy model we have assumed all the energy emitted from the surface is absorbed in L1, despite the fact that the atmospheric window actually allows about 70% of all surface radiated longwave energy to pass through the atmosphere unabsorbed and directly into space.  So Sp really equals 0.3 and the first five emissions to space really equal 0.2288.  In this simple model with no thermalization of the absorbing molecule and no significant half-life before it re-emits energy as radiation, sending more than 3/4ths of the surface emitted energy into space takes about the time it would take radiation traveling at the speed of light, 3 x 10⁸ m/s, to travel through the troposphere about 11,000 m high five times.  That time is 3.7 x 10^-5 s.  In comparison, it takes hours for the alternative heat dissipation process of convection currents to raise surface energy to the top of the troposphere where it can be directly emitted to space as radiation.

Of course, the catastrophic man-made global warming argument does not emphasize the speed with which energy is emitted to space by a radiation-centric model.  They emphasize the added time radiation energy spends near the surface because of their isotropic emission model in comparison to the time it would take to go directly from the surface to space if there were no infrared-active molecules.  So what enhancement of radiative energy dwell time are they getting?  With this two atmospheric absorption layer model it would be that the sum of radiation back to the surface from L1, Su, is

This means that the energy dwell time in the lower troposphere has been doubled by this two-layer 100% radiative energy loss model with isotropic emission.  But keeping about 76% of this energy around for about 3.7 x 10^-5 s is not such a big deal.

What if we put three atmospheric absorbing layers into the model?  Then I find that the series of emissions to space from the top layer L3 is

Sp = 1/8 + 1/8 + 7/64 + 3/32 + 41/512 + 35/512 + 239/4096 + 577/16384 + ….

These first eight terms sum to 0.695129, so with eight emissions from L3 to space 69.5% of the unit of energy emitted from the surface has been lost to space.  Consequently, 69.5% of the energy is lost in about 8 (11,000 m) / 3 x 10⁸ m/s or 2.9 x 10^-4 s.

Su = ½ + 3/8 + 5/16 + 17/64 + 29/128 + 99/512 + 169/1024 + 577/4096 + 1731/16384 + …..

The first eight terms of Su sum to 2.01898.  So when Sp is a bit over 2/3 after eight terms and must sum to one, Su is a bit over 2 and seems most likely to sum to about 3.

So, let us make a leap here and assume that with an atmospheric absorption model of n layers with isotropic emission, Su will sum to about n.  [If someone has the time to work this series out to more terms or can find a way to solve it exactly, I would enjoy seeing the result.]

Alright now, let us assume that we have 100 absorption layers in our atmosphere, corresponding to an absorption distance of about 110 m.  Most of the radiation energy emitted from the surface will find its way to space in less time than 100 (11,000) / 3 x 10⁸ = 3.7 x 10^-3 s.  If you have 10,000 atmospheric absorption layers (an absorption distance of about 1.1 m), the time is then 0.37 s.  But the alternative means of removing that energy to space is convection and that takes hours to do the job.

In fact, radiation between the layers only occurs long after an absorption event during which time many, many collisions with other molecules would occur and the absorbing molecule would give up virtually all the energy it had absorbed from radiation from another layer to the 2500 times as plentiful non-greenhouse gas molecules in the air.  It is a comparatively very long time before the infrared-active molecule emits radiation again.  In the meantime, it is only 1/2500 of the molecules moving energy as part of a convection current.  Yet insofar as these infrared-active molecules do emit radiation, they are acting to speed up the emission of surface energy to space.  They are therefore acting to cool the atmosphere from top to bottom of the troposphere compared to the convection energy transport mechanism.

The lesson here is that the very first absorption event of thermal radiation from the surface in the atmosphere is very important because it puts the transfer of that energy into the hands of a much slower convection cooling process than is that of radiation.  However, whatever further thermal radiation events occur in the atmosphere simply speed up the loss rate of energy to space compared to the rate due to convection.  The addition of further infrared-active gases to the atmosphere causes there to be more absorption layers in the model.  If the atmospheric load of infrared absorption gases was so high that the mean free path length for absorption of their emissions was as short as 1 meter, then the time to dissipate most of the energy to space would still be less than a second, while the time it takes for the alternative energy transport by convection is still hours.

Do additional greenhouse gases warm or cool the Earth?  The addition of a gas in just enough concentration that there is absorption by that gas in the lower troposphere once which would not otherwise have occurred at a given wavelength slows the rate of radiative heat loss and may be regarded as effectively warming the Earth.  It does this by converting the cooling mechanism from rapid radiative cooling to that of slow convection cooling.  However, once that threshold concentration is exceeded for a given wavelength, additions of that gas simply cool the atmosphere more quickly than would the alternative of convection currents.  At such an above threshold concentration, that gas can be regarded as cooling the Earth faster compared to the rate it would cool without its addition.

The lapse rate is the temperature gradient with altitude in the troposphere.  At normal levels of humidity, the adiabatic lapse rate is less than the dry lapse rate is.  This tells us that at normal water vapor concentrations, the water vapor concentration is already high enough to produce a cooling effect on the surface and lower troposphere temperatures.  Water vapor does this with the cooling effect at the surface as liquid water becomes water vapor and then the water vapor rises with convection until it reaches an altitude at which it condenses and releases energy.  At the warm surface it cools with evaporation and at the cooler altitudes it warms by condensing.  The evaporation process increases the water molecule’s kinetic energy, including its vibrational modes, and the molecule carries that energy upward by convection and then releases the kinetic energy of evaporation as it condenses to liquid or solid form.   Each water molecule carries more energy per molecule at a given temperature than can a nitrogen or oxygen molecule.  Thus, as they rise with convection, they are transporting more energy upward per molecule than are the dry air molecules in the same convection current.  In addition, the water molecule is radiating energy to the layer of air above it, which is usually cooler and able to absorb that radiated energy if it also has water vapor molecules in it or sometimes if it has carbon dioxide in it. 

Unlike water vapor molecules, a carbon dioxide molecule carries less energy at a given temperature than do the nitrogen and oxygen molecules with which it shares a convection current.  This means an added CO₂ molecule causes a convection current to become a less effective cooling mechanism.  However, it retains the ability to warm the air layer above it throughout the troposphere as it cools its local surroundings by radiation to the layer above it.  In addition, more CO₂ in the upper troposphere and in the stratosphere means more molecules radiating energy directly to space.  There is good evidence that the addition of more hot molecules of CO₂ in the stratosphere has resulted in a measured  cooling of the stratosphere, as would be expected because these hot molecules are effective radiators.  I believe the concentration of carbon dioxide in the atmosphere is already high enough that additions of CO₂ are cooling the troposphere as well or at least counterbalancing the mild warming effect of additional carbon dioxide molecules to a great degree.

Whether I am right or not about this, the claim that a very small warming effect by additional CO₂ will be amplified by a greater warming effect by increased water vapor (a positive feedback) is surely false.  The fact that the wet adiabatic lapse rate is less than the dry adiabatic lapse rate makes it clear that the claim of a positive feedback is wrong.  The IPCC and other alarmists depend upon this false claim of a positive feedback by water vapor to make it appear possible that additional carbon dioxide will cause significant harm.  The reality is that additional carbon dioxide has no net significant effect on temperatures at the Earth’s surface or in the lower troposphere.

Meanwhile, additional carbon dioxide in the atmosphere provides plants with the means for easier growth.  With a growing human population, this is very helpful in producing the additional food we need to produce.  This should be a factor in reducing human anxiety for the future.  Of course, I understand that some people just have to have something to worry about.  I suggest you worry, if you must, about another ice age which additional carbon dioxide in our atmosphere cannot prevent.  Or, you might worry about an asteroid  striking the Earth.  But it is even more foolish to worry about problems created by more CO₂ in the atmosphere.