Posts Evaluating Earth Energy Budget Problems

Here is a list of some of the posts in which I discuss some of the many problems in the Earth Energy Budgets put forth by the advocates of catastrophic man-made global warming:

Using Heat Transport Powers of the NASA Earth Energy Budget to Prove that Carbon Dioxide has an Insignificant Effect on Surface Temperatures, 15 June 2018

A Summary of Some of the Physics Errors of the NASA Earth Energy Budget, 8 June 2018

Water Vapor and Gravity Act Together to Warm the Earth, 19 June 2017

The Settled Science of Catastrophic Man-Made Global Warming Violates the Laws of Physics, 10 June 2017

mgh, Not Just Greenhouse Gases, Provides a Warm Earth, 23 May 2017

Greenhouse Gases Warmed the Earth Somewhat, but Additions Now Cool the Earth, 27 July 2015

Why Greenhouse Gas Theory is Wrong -- An Examination of the Theoretical Basis, 7 March 2015

Back-Radiation Insignificance for the Equilibrium Surface Temperature, 14 March 2014

The Stefan-Boltzmann Law at a Non-Vacuum Interface: Misuse by Global Warming Alarmists, 6 April 2013

The Unsettled Earth Energy Budget, 24 February 2013

Back-Radiation and the Highly Fallacious Kiehl-Trenberth Energy Budget, 10 July 2012

NASA Finally Produces A Realistic Energy Budget for the Earth, 26 May 2011

Back-Radiation and the Highly Fallacious Kiehl-Trenberth Energy Budget

The usual greenhouse gas calculation such as is offered up by the various Trenberth diagrams is highly dominated by the radiative transport of energy in the Earth’s atmosphere.  It leads to an unreasonable result for the situation I am going to describe here.

But first, my approach to the back-radiation problem was always to show that it was unrealistic on many levels.  One cannot make a net gain of heat energy in a surface by returning a portion of the energy lost from a surface to it.  This is especially true of any process that is said to lose half of the surface emitted energy to space immediately and to only re-absorb some part of half of the energy that was returned to the surface.  I went further and explained that only a small fraction of the energy emitted as radiation from the surface was re-emitted as radiation from an absorbing water or CO2 molecule.  With just these considerations alone, the upper limit on once radiated energy from the surface which is re-absorbed by the surface would be:

(0.5) f,

where the 0.5 is the half of any IR absorbing molecule radiation which was not radiated into space and is radiated toward the surface.  The fraction f is the fraction of the IR energy lost by the surface and absorbed by an IR absorbing molecule which is re-emitted before collisions have dissipated the energy absorbed.  The fraction f is much less than one because most often the IR absorbing molecule undergoes collisions with other molecules and transfers much of the IR absorbed energy to other molecules before re-emission occurs.  These other molecules are rarely water or CO2 or other IR emitting molecules, so that energy is not returned to the surface as IR radiation.  Thus much of the surface emitted IR radiation is dissipated to the 99.97% of the atmosphere molecules which are nitrogen, oxygen, or argon and are not IR emitters.

Let us estimate the value of f.  At sea level, the mean gas velocity is 459 m/s, the mean free path or distance between collisions is only 6.6 x 10^-8 m or 66 nm, and the collision frequency is 6.9 billion/s.  At an altitude of about 4000 m, the radiative transfer of energy competes about evenly with transfer by collisions.  At 4000 m altitude, the frequency of gas molecule collisions is about 4.4 billion/s.  We can use the equivalency of energy transfer by radiation and gas molecule collisions at the 4000 meter altitude to estimate the fraction of energy transfer by radiation of the total of energy transferred by radiation plus gas molecule collisions. At sea level, energy transfer by radiation is equivalent to about 4.4 x 109 collisions per second, so the fraction of energy transferred by radiation is about 4.4/(4.4 + 6.9) = 0.39 of the total by gas molecule collisions and radiation.  This suggests that about 1.5 times as much energy is transferred by gas collisions as by radiation at sea level.

So at this point, the upper limit on IR radiation emitted from the surface which can be returned to the surface and absorbed by it is about:

(0.5) (0.39) = 0.195

It also has to be remembered that this is the upper limit for that portion of the IR radiation from the surface which can be absorbed by an IR-absorbing molecule such as water or CO2.  Much of the surface-emitted IR radiation is of such wavelengths that no IR-absorbing molecule can absorb it in the first place.  If one takes this fact into account, the 0.195 upper limit is a hugely generous upper limit.

But, it does not follow that simply because this much back-radiated IR is incident upon the surface that it will be absorbed by the surface.  During the roughly 8 hours of a day when the surface is warming under increasing sunlight and with a 2-hour lag for warming the ground, water, or air in a vicinity, none of this radiation may be absorbed, except when a cloud casts a shadow on a part of the area or at such points under the shadow of a tree of some such object.  The absorbing ground has to be cooler than the ground from which the photon was emitted and subsequently absorbed by an IR-absorbing molecule in the atmosphere.  During the remaining cooling hours of the day, roughly 16 hours, the surface is more likely to absorb such back-radiated energy.  As a mean value for re-absorption of IR back-radiation over the daily cycle, the value of 0.95 is often used.  I believe that value is much too high.

Let us look at one of the Kiehl-Trenberth energy budgets for a moment:

According to this diagram, 67/ 342 = 0.196 of the incoming solar radiation is absorbed by the atmosphere.  77/342 = 0.225 is reflected by clouds and aerosols.  The surface reflects 30/(30 + 168) = 0.152 of the solar radiation incident upon the surface.  Thermals cooling the surface dissipate 24/168 = 0.143 of this incident radiation.  Let us note that the surface emits a flux of 390 W/m^2 of IR radiation and according to this diagram the atmosphere has absorbed a highly efficient 324 W/m^2 of this or 83.1% of this.  Yet, above I showed in very simple terms that 19.5% is more than the upper limit by far of the amount of IR radiation that the atmosphere can return, which is not from IR-absorbing gas molecules that are substantially cooler than the Earth's surface.

Now let us consider a midday calculation of the surface and do so where there is no cloud cover and where it is so dry that there is no evapotranspiration.  What kind of surface temperature will we have.  Since there are no clouds and I think clouds are the better part of the summed cloud and aerosol effect, let us assume the aerosol effect alone is 0.08.  The midday radiation incident on the upper atmosphere is 1367 W/m^2.  Note that at midday, the incident radiation path length through the atmosphere is shorter than it is for the average daily values normally used, so losses in the atmosphere should be lower than these numbers.  I will use them nonetheless.

The radiation upon the ground is then:

(1 - 0.196) (1- 0.08) (1 - 0.152) (1 - 0.143) 1367 W/m^2 = 734.8 W/m^2

The ground temperature is then found from:

734.8 W/m^2 = ε σ T^4,  ε = 0.95

T = 341.8 K = 68.6̊C = 155.5̊F

But if you believe that the upper limit amount of back-radiation is 95% absorbed by the surface, then the incident radiation is:

(734.8 W/m^2) (1 + 0.195(0.95)) = 870.9 W/m^2

T = 356.6 K = 83.4̊C = 182.2̊F

Now this is clearly much too hot and implies that even the addition of this upper limit of back-radiation which is much smaller than the back-radiation in the Kiehl-Trenberth diagram and energy budget is not physical.  In reality, air convection or thermal effects are more important in cooling the surface than the Kiehl-Trenberth energy budget allows for.  The same is true of water evaporation, transport, and condensation effects.

Note that I did not follow the Kiehl-Trenberth diagram in subtracting the emitted radiation that corresponds to the Earth’s surface temperature.  They subtract 390 W/m^2 and add back in 324 W/m^2 of back-radiation for their mean daily calculation.  This would imply that if we had no atmosphere with IR absorbing gases in it, then there would be no back-radiation, so the total energy budget would look like:

(168 - 24 - 78 - 390) W/m^2 = -324 W/m^2,

which is nonsense.  Of course you might say that the thermals would be different and if I have no absorbing IR gases, then I certainly do not have water evaporation and movement.  This perhaps is a muddled situation.

So let us consider the equivalent calculation technique for an isolated black body radiator in space with incident energy flux of Ii and an emitted radiation of Ie.  But Ii = Ie, so then this approach would have us fallaciously conclude that

Ii - Ie = 0 = σ T^4 and T = 0 K.

The temperature of the isolated black (or gray) body is determined by the incident radiation on it.  The basic approach of the Kiehl-Trenberth diagram to radiation is nonsense.  The energy budget is a farce based on bad physics.  Indeed, when challenged on this issue, many proponents of man-made catastrophic global warming back down and say that the General Circulation Models are calculated primarily as air and water vapor circulation models and are not really consistent with the several variations of the Kiehl-Trenberth Energy Budget.  Yet it is such fallacious energy budgets that the public has been fed as the basis for the claim that there are substantial effects on the Earth's surface temperature due to man's emissions of CO2.  Government websites have been full of these energy budgets, as have college classes.  For their part, the range of results in the GCM computer models is too large to be consistent with the idea that climate science is settled and everyone agrees that it is understood.

NASA Finally Produces A Realistic Energy Budget for the Earth

The Atmospheric Science Data Center of NASA has finally produced a realistic energy budget for the Earth.  [The link no longer works.  It is not surprising that the diagram fatal to the back radiation model of man-made catastrophic global warming due to back radiation acting on greenhouse gases is no longer publicized by NASA.  But I did find it in this educational product pdf file.  23 Feb 2013]  There is no back-radiation in this diagram due to greenhouse gases, or really due to infra-red absorbing gases, such as water vapor, CO2, and methane.  The new NASA energy budget is:

Note how remarkably different this is from the Kiehl-Trenberth diagrams NASA has previously been using:

I have discredited the Kiehl-Trenberth diagram in this post.

While the new NASA energy budget is much improved and has no back-radiation component, the web site I linked to above still talks briefly about greenhouse gases and some back-radiation effect.  As I have noted, any such effect is small and can only happen in a way to briefly slow the cooling of the Earth as solar insolation is being reduced from its maximum effect in the afternoon of a day.  Even then, infra-red absorption of incoming radiation from the sun may be greater than any back-radiation effect until evening.  The overall effect of IR-absorbing gases is a cooling effect as I have previously claimed it was.

Let us consider the new NASA Earth Energy Budget numbers.  The first thing to note is that the infra-red radiation from the ground is only 21% of the incoming radiation from the sun.  In the Kiehl-Trenberth diagram, the energy flux of out-going infra-red radiation (396 W/m^2) is actually greater than the total incoming solar radiation flux (341 W/m^2) and much, much greater than the energy absorbed by the surface from the sun (161 W/m^2).  This is patent nonsense.  Energy must be conserved and it is in the new diagram.

As I have claimed, the net effect of infra-red absorbing or greenhouse gases is that they absorb more energy from the incoming solar radiation through the course of the day than they absorb from the out-going infra-red radiation from the surface of the Earth.  In the new NASA diagram, the total radiation absorbed by the infra-red absorbing gases is 16% + 3% (clouds) = 19%.  Of the 21% of infra-red radiation emitted from the surface, only 15% is absorbed in the atmosphere.  Thus, the net absorption of incoming solar radiation at 19%, is 1.27 times greater than the net absorption of energy emitted by radiation from the ground.  This means the net cooling effect is at least 1.27 times greater than any warming effect could be.

Another point of great interest is that the new NASA diagram gets the fraction of the cooling of the surface of the Earth by infra-red radiation about right.  The fraction of surface cooling by infra-red radiation from the surface is 21% / 51% = 0.41.  This agrees with some calculations I have done based on molecular collision frequencies at the altitude in the atmosphere at which radiation competes evenly with molecular transport or movement.  The temperature of the Earth seen from space as a black body radiator is 255 K.  This is the temperature of the standard atmosphere at an altitude of 5 Km.  The altitude at which radiation cooling in the atmosphere is equal to the cooling caused by thermals, conduction, and the transport of water vapor is then a bit lower and proves to be about 4 Km.

Water vapor is the best long wavelength IR absorber and it is the best emitter of IR energy, but before it can commonly emit the energy it has absorbed from IR radiation, even it will likely suffer numerous gas collisions with much of its excess molecular energy being transferred in those collisions to the molecules which collide with the water molecule.  Nitrogen molecules are the most likely molecules to take up much of the energy from the water molecule, since nitrogen is 78.08% of the atmosphere.  Oxygen molecules are the next most likely colliders at 20.95% and then argon atoms at 0.93%.  Together, these three gases account for 99.97% of the U.S. Standard Atmosphere.  None of these gas molecules are very efficient IR absorbers in the long wavelength spectrum.

At sea level, the mean gas velocity is 459 m/s, the mean free path or distance between collisions is only 6.6 x 10^-8 m or 66 nm, and the collision frequency is 6.9 billion/s.  At an altitude of about 4000 m, the radiative transfer of energy competes about evenly with transfer by collisions.  At that altitude, the frequency of gas molecule collisions is about 4.4 billion/s.  We can use the equivalency of energy transfer by radiation and gas molecule collisions at the 4000 meter altitude to estimate the fraction of energy transfer by radiation of the total of energy transferred by radiation plus gas molecule collisions. At sea level, energy transfer by radiation is equivalent to about 4.4 x 10⁹ collisions per second, so the fraction of energy transferred by radiation is about 4.4/(4.4 + 6.9) = 0.39 of the total by gas molecule collisions and radiation.  This suggests that about 1.5 times as much energy is transferred by gas collisions as by radiation at sea level.  Note that this number is in good agreement with the fraction of the energy 0.41 given in the new NASA diagram.

This phenomenal number of atmospheric molecular collisions spreads the IR energy absorbed by a water molecule or a CO₂ molecule from the ground long wavelength IR emissions to the dominant nitrogen and oxygen molecules very, very quickly.  At an altitude of 5 km, the collision frequency is still 3.9 billion/s and at 10 km altitude it is 2.1 billion/s.  If a water molecule is to radiate energy away as IR emissions, it must do so very quickly!  If it were able to emit IR very quickly, then the atmosphere would cool down very quickly and effectively at night.  Indeed, cooling at high elevations in mountains by radiative cooling is more effective than cooling from sea level because less of the radiative energy of the ground is spread to many nitrogen and oxygen gas molecules which then tend to hold the energy near the ground.

Back radiation to the ground from an infra-red absorbing molecule such as water or CO2 is thus not a likely event.  To this, we must add the very important caveat that the Earth's surface can only re-absorb that energy if it has cooled since the emission of the radiation occurred.  Given the very short time scale between emission of radiation from the Earth and the re-emission from an atmospheric molecule such as CO2 or water, any such cooling is trivial.  Radiation from molecules in equilibrium with the cooler temperatures found as one goes higher into the atmosphere will not be absorbed at all, since infra-red radiation only flows from warmer to cooler bodies.  The reverse does not happen.  This is a law of thermodynamics often violated by greenhouse gas global warming alarmists.  Thus, any back-radiation effects were always trivial and more than compensated by the cooling effect of infra-red gases absorbing solar radiation before it could reach the surface of the Earth.