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

One of the significant errors commonly made by the advocates of catastrophic man-made global warming due to CO₂ emissions is the claim by the settled science proclaimers that radiation from a non-vacuum interface is the same as radiation from a surface into a vacuum.  This error in the physics of radiation from the Earth’s surface results in an exaggeration of the cooling radiation emitted from the Earth’s surface and contributes to them positing a hugely larger back-radiation from greenhouse gases than can actually occur.

I have previously pointed out that the Stefan-Boltzmann Law actually only tells us the amount of radiation emitted by a surface into a vacuum.  A surface in contact with another material will lose energy by other mechanisms, so one must apply the law of Conservation of Energy to determine the actual amount of radiation in many cases of material contact across an interface.  In the case of the Earth’s surface, water is evaporated at the surface with a very substantial cooling effect.  In addition, air molecules strike the surface and carry away heat gained in collisions with the surface.  Despite these obvious problems with an unchanged surface emission of radiant energy into the atmosphere compared to that into a vacuum, the settled science proclaimers have in many cases steadfastly said that I am wrong.  OK, so I will try to explain this in greater detail in this post.

Atoms in solid materials such as in soil and rock, are held at distances from one another which are determined by a minimum in the potential energy.  The atoms can only be forced closer with the expenditure of energy and they can only be pulled further apart with the expenditure of energy.  An electron in orbit about a nucleus will also have motion constrained by a potential energy well.  The greater the temperature, the more an atom may move near the potential energy minimum and the more the electron can move in the nucleus-electron potential well.  In both cases, positive and negative charges will move with respect to one another.  When positive and negative charges are close to one another, but have offset centers of charge, they form a dipole.  Because the displacement movements of the charges in the dipoles are small for the temperatures near the Earth’s surface, the results are dipole charges with an oscillating distance between them similar to a mass hanging from a spring in small motion.  These are harmonic oscillators and they emit radiant energy.  While the interatomic potential energy wells in a liquid are broader than those in a solid material, the same principle applies to liquids.  Of course, in either a solid or a liquid, atoms have several nearest neighbors or near neighbors.  Multiple harmonic oscillators are interacting.

If a harmonic oscillator in a vacuum is set into motion by a heating process and the heat source is removed, the harmonic oscillation will lose strength as it emits energy into a strengthening radiant energy field.  Conversely, an increase in the harmonic oscillation caused by an electromagnetic field, such as the solar insolation acting on the Earth's surface, will decrease the energy in the radiation field.  A cooling surface radiating energy will have decreased harmonic oscillator displacements as it pours energy into the electromagnetic field.  A surface near 300K will generate infra-red and microwave radiation, though almost all of the energy given off will be in the infra-red radiation range.  The generation of a radiant energy field decreases the kinetic energy of the harmonic oscillators in the surface.  There is Conservation of Energy between the harmonic oscillators and the electromagnetic field which is generated.

The Stefan-Boltzmann Law tells us how much energy is radiated per unit time into the electromagnetic field of the vacuum:

P = ε σ A T⁴,

where P is the power, ε is the emissivity and characteristic of the surface, σ is the Stefan-Boltzmann constant, A is the surface area of the radiating material, and T is its temperature in Kelvin.

We must remember however that the radiated energy comes from harmonic oscillators.  If the surface is a water surface or if it is a soil or a plant with water content and water is evaporated from the surface, we must remember that the energy required to change water from its liquid to its vapor form has to come from somewhere.  The Earth’s surface does have considerable water evaporating from it and the latent heat of vaporization for water is very high.  Where does this energy come from?  Well it comes from the kinetic energy of the oscillating dipoles at temperatures near the average Earth temperature of 288K.  As the warm surface materials evaporate water, their harmonic oscillators lose kinetic energy and settle more towards their potential minima except insofar as the energy is replaced by more solar insolation or by heat flow from the subsurface.  The oscillation displacements decrease.  There is a conservation of energy between the harmonic oscillators and the energy used to evaporate the water.  The same is true when the harmonic oscillators warm air molecules that strike the Earth’s surface.  Those air molecules take away some of the kinetic energy of the harmonic oscillators.

Consequently, the harmonic oscillators that generate radiation into vacuum will not be able to generate as much radiant energy into the atmosphere.  The presence of contacting liquid water and air molecule collisions with the surface remove energy from the harmonic oscillators that generate the radiation field.  Consequently, the amount of infra-red and microwave energy emitted from the surface will be less than if that surface were radiating into vacuum.  It has to be so because energy is conserved.

This is why it is clear that the Stefan-Boltzmann Law tells us the maximum energy that can be obtained from a warm surface of material.  At a vacuum interface, that energy given off will be entirely radiant energy and it will all go into strengthening the electromagnetic field.  The Earth’s surface however is taking the same kinetic energy from its many harmonic oscillators and it is partitioning that energy among the processes of generating water vapor, warming colliding air molecules, and emitting radiant energy.  The amplitude of the oscillation in the harmonic oscillators decreases as they pour energy into these three loss mechanisms.  Energy is thus conserved.

The Kiehl-Trenberth Earth Energy Budget used so prominently in the UN IPCC 4th Report of 2007 made the mistake of not adjusting the Earth's surface radiation downward due to the evaporation of water and the warming of air.  Here is that diagram:

It is claimed in this diagram that the Earth's surface at 288K emits 390 W/m² or 114% of the average power incident at the top of the atmosphere and 2.32 times the power absorbed by the surface from solar insolation.  The 390 W/m² of surface radiated energy assumes that the Earth’s surface is a black body radiator emitting the same energy it would into vacuum into the atmosphere.  As I have explained in my post The Earth Surface Temperature without Greenhouse Gases: The Shade Effect of Infra-Red Active Gases, the Earth’s surface is not a black body radiator.  It does not have an emissivity of one as a black body radiator does.  It has an emissivity of less than 0.5.  But even if it were a black body radiator, energy conservation would require that the emitted radiant energy be (390 – 78 – 24) W/m² or 288 W/m² due to subtracting the energy put into evaporation and thermals.  This is 112 W/m² less than they claim is emitted from the surface.

They then make the further mistake of believing that most of that exaggerated surface emitted radiant energy is returned to the surface by greenhouse gases.  This makes up much of the 324 W/m² they claim is absorbed by the surface after it had been absorbed in the atmosphere first.  The back radiation energy is thus exaggerated hugely by a combination of errors.  Among the errors are:

• The belief that the Earth’s surface is a black body radiator with emissivity 1.

• The violation of the Conservation of Energy by failure to subtract the energy used to cause evaporation and to generate thermals from the energy that would be emitted as radiant energy into vacuum.

• The failure to understand the consequences of the high gas molecule collision rates near the surface and the very short mean free path of infra-red radiation which can be absorbed by water vapor and carbon dioxide, as well as their rarity among air molecules.  In the lower troposphere, energy is almost entirely transported upward.

The most recent Earth Energy Budget posted by NASA makes the same errors:

 

This is an incredible comedy of errors for science that has been funded by about $140 billion of hard-earned taxpayer money.  It is a comedy of errors with very tragic consequences.  Obama and his Democrat Socialist Party are still calling for the destruction of the American economy and the lowering of our standard of living for the supposed purpose of saving the world from catastrophic man-made global warming due to the use of fossil fuels.  This disastrous crusade is pursued in the name of this childishly wrong "settled" science.  A great many scientists are eagerly participating in this pandering for the goodies handed out by a government eager to exercise still more power while pretending to protect us from a fictional catastrophe.  We are ruled by pimps and fools.