Simple Explanation of Why Greenhouse Gases Do Not Warm the Earth's Surface

This is a simple explanation why an essential claim of greenhouse gas theory or the catastrophic man-made global warming hypothesis is very wrong.  That theory believes that back-radiation from the atmosphere to the surface is about the same as the total solar radiation incident upon the top of the Earth's atmosphere.  For a much longer exposition of many of the problems with this theory you might examine my earlier and much-updated post

AGW Theory: Back Radiation Insignificant for Surface Temperature

This post is comparatively very short and will only discuss a simple explanation of the cornerstone argument of huge back-radiation to the surface.  Indeed, I will show why that back-radiation is virtually zero.  Because it is nearly zero, the catastrophic man-made global warming hypothesis is shown to be nonsense.  There is no excuse for governments acting to force individuals and companies to reduce their carbon dioxide emissions.  Doing so is very expensive, causes energy reliability problems, and puts people out of work.  Causing these miseries in the name of a failed hypothesis is a very serious injustice to mankind.

Let us examine a current NASA depiction of the huge role the catastrophic AGW hypothesis gives to back-radiation:

Now notice that the emitted surface radiation is 117% of the top of the atmosphere average radiation from the sun and back-radiation is 100% of the top of the atmosphere incident solar radiation.  Notice also that only 48% of the total solar radiation was ever absorbed by the Earth's surface.  Of the 48% of the total solar incident energy absorbed by the surface, 25% is lost by evaporation of water and 5% by convection loss, leaving only 18% to be emitted by radiation.  Now the proponents of the large greenhouse warming effect claim that the 18% is just a net difference between a larger surface radiation emission rate and a large back-radiation rate.  In this case the difference of 117% - 100% = 17%, which is about 18%.  We are going to evaluate this claim.

Consider the surface of the Earth for a moment.  The density of atoms per cubic meter in the surface is about 1 gram per cubic centimeter for the water that covers about 71% of the Earth's surface and even greater for land materials or if considering the salts in the oceans.  Expressed as a density per cubic meter of water, this is 1000 kg per cubic meter.   A cubic meter of water has 3.34 x 10²⁸ water molecules.  Infra-red emission and absorption in a surface occurs in the outer 2 micrometers of the surface.  Consequently, there are about 6.7 x 10²² water molecules emitting infra-red at most from the surface.  When infra-red radiation is incident upon the surface, these are the same water molecules that would be able to absorb that radiation.

The density of the atmosphere near the surface at sea level is 1.225 kg per cubic meter in the U.S. Standard Atmosphere at a temperature of 288.15K.  Adding water vapor very slightly decreases that density.  The number of molecules per cubic meter at sea level in the U.S. Standard Atmosphere is 2.55 x 10²⁵/m³.  The important infra-red active gas near the surface is water vapor and its density per cubic meter is commonly between 10 g/kg to 14 g/kg of air as shown in Fig. 1. below.  At a specific humidity of 12 g/kg, the number of water molecules/m³ of air is about 4.9 x 10²³.  Therefore, there are more water molecules in the first cubic meter of air above 1 m² of water surface emitters by a factor of 7.3.  This should mean that radiation that can be absorbed by water vapor will be absorbed in the first meter of air above the surface at a humidity near the Earth average humidity.  There is a report that the average mean free length for water vapor absorption in the atmosphere is as long as 8 m.  If it actually is that long, then the result I get below on the amount of back-radiation can be multiplied by a factor of 8.  That number will still be most unimpressive.

Fig. 1. The average specific humidity of air is shown as a function of latitude.  The specific humidity is the weight of water in grams in one kilogram of air.

In comparison, at 400 ppm of carbon dioxide, there are 1.0 x 10²² molecules of CO₂ /m³.  There are 49 water vapor molecules for every carbon dioxide molecule at a specific humidity of 12 g/kg.  In addition, the emissivity of CO₂ molecules is less than 40% that of water vapor molecules.  Since many of the carbon dioxide molecule absorption peaks are largely or mostly already absorbed by overlapping water vapor absorption lines, the effect of CO₂  near the surface is very minimal compared to that of water vapor.  This is less true at altitudes above 4 km where the concentration of water vapor drops greatly and the carbon dioxide remains in proportion to nitrogen and oxygen, but such altitudes have nothing to do with the back-radiation to the surface issue.

Returning to the back-radiation caused by water vapor, we find that the temperature difference over the 1 meter absorption range according to the U.S. Standard Atmosphere is only about 0.0065 K/m.  For humid air, the temperature gradient is even less.  So if the water surface and the water vapor in the first meter of air above the surface are treated as gray bodies, we have a power transfer from the surface to the water vapor in the atmosphere of P_W

P_W = σ (ε_s T_s⁴ – ε_a T_a⁴) , 

for the surface s and the atmosphere water vapor a and if we take the emissivities to be a high value of 0.95 often used by the catastrophic man-made global warming promoters, this is equal to 0.033 W/m².

This is actually an over-estimate because we have ignored the fact that some of the heat energy in the surface is used to evaporate water and some is lost to the air by means of air molecule collisions with the surface.  Much of the solar radiation absorbed by the surface is actually transferred out of the surface by evaporation and by conduction.  Of the 48% of solar radiation absorbed by the surface according to the above NASA schematic, a total of 30% is removed from the Earth's surface by evaporation and convection/conduction.  This leaves only about 18% to be radiated away from the surface.  So if the Earth's surface was an interface directly with vacuum, the 48% of solar radiation absorbed would be radiated in its entirety.  But other processes, evaporation and conduction, remove most of this energy, leaving a fraction of only 18%/48% = 0.375 of the total energy flux to be removed by radiation.

But water vapor does not absorb infra-red across the entire surface emission spectrum.  It actually absorbs only about 65% of it.  At least 28% of the surface emission is into the atmospheric window where no atmospheric gases absorb the infra-red radiation.  See the transmission spectrum of the Earth's infra-red surface emissions taken by Nimbus IV in 1970 near Guam below:

This points out one of many flaws in the NASA depiction of the Earth Energy Budget above.  Taking the two atmospheric windows at higher wave numbers than the peak of the Earth emission spectrum, the infra-red radiated into them is about 28%.  28% of the 117% surface infra-red emission claimed by NASA is 33%, not the 12% quantity their schematic assigns to the atmospheric window radiation lost directly into space from the surface.  The direct loss to space of surface emission with the NASA depicted 117% surface emission should be 2.75 times what NASA puts in its Earth Energy Budget schematic diagram at the start of this article.  This inconsistency suggests that the surface emission radiation is actually much less than the 117% claimed.  If that emission is much less, then the back-radiation would also have to be much less.

So the actual water vapor absorbed infra-red radiation is only about 

P_W =  (0.375) (0.65) (0.033 W/m²) = 0.0080 W/m².

But because of the 6.9 x 10⁹ collisions/s in air near sea level, most of this energy is transferred to non-radiating nitrogen, oxygen, and argon.  Only about 20% is re-radiated and half of that is radiated toward space.  Consequently, the total back-radiation which can be absorbed by the surface, P_B, is about

P_B = (0.2) (.5) (0.0080 W/m²) = 0.00080 W/m²

Thus, the absorbed back-radiation has an upper limit of about 0.00023% of the average solar insolation at the top of the atmosphere (342 W/m²)!  For all intents and purposes, the absorbed back-radiation is zero.  This value should be multiplied by the mean free path for absorption by water vapor, which may be as large as 8 m on average.

I just proceeded down the true path of analysis dictated by the use of Occam's Razor.  Let us assume that NASA is right that 117% of the solar incident radiation at the top of the atmosphere actually is emitted from the surface due to some unknown extra input energy that lifts the 48% of the solar absorbed energy flux at the surface by 117% + 25% + 5% - 48% = 99% so that one can have 117% radiation, 25% evaporation, and 5% conduction cooling mechanisms at work.  Is it possible for that 99% warming source to be back-radiation?

The 117% radiation is greater than the 18% upon which I performed the Occam's Razor version of the calculation.  In fact, the back-radiation per meter of mean free path is then 117% / 18% = 6.5 times greater than the value I calculated.  So, if we assume that the average water vapor mean free path is 8m, we get 8 (6.5) (0.00023%) = 0.012%, which does not look at all like the 100% claimed by NASA.


Because back-radiation to the surface is insignificant compared to the claims made by the proponents of catastrophic man-made global warming, the mechanism upon which that theory stands is wrong.  Indeed, adding carbon dioxide to the atmosphere will actually cause more incoming solar radiation to be absorbed by the atmosphere before it reaches the surface.  This results in a cooling of the surface.  In addition, more CO2 in the atmosphere near the surface will also cause the temperature gradient in the atmosphere to become slightly smaller, just as infra-red absorbing water vapor makes it smaller.  This is because radiation transport effects operate at the speed of light, which is faster than evaporation/condensation transport or conduction/convection transport of energy.  Since all remove energy from the surface, they are all cooling effects.  As a result, adding CO2 to the atmosphere actually causes a very slight cooling of the surface, contrary to the claims of a substantial warming effect.

The primary sources of CO2 in the atmosphere are warming oceans, decaying plant life, and heat vents and volcanic emissions.  Since CO2 in the atmosphere creates a slight cooling of the surface, it acts as a negative feedback to the warming oceans that cause it to increase and it slightly cools the decaying plants to slow down the further generation of CO2 from that source.  Of course volcanic and heat vent sources of CO2 are also providing heating, so carbon dioxide as an surface coolant acts to stabilize the Earth's temperature much as water vapor does.  It has negative feedback effects.  It is no more subject to the sort of tipping point catastrophes that global warming alarmists put out than is water vapor, though its effects are much, much weaker.

This article was last updated on 29 July 2014. 

[The fact that back-radiation itself is insignificant in explaining the 33C warming of the surface claimed by the proponents of catastrophic AGW is not at all to say that the effects of water vapor on the surface temperature are without significance.  During the day the evaporation of water is an important surface cooling effect and during the night its condensation can be an important warming effect.  During the day, the infra-red active gases absorb a portion of the incoming solar radiation and prevent it from being absorbed by the surface.  At night, fog and clouds scatter, reflect, and absorb the infra-red radiation emitted from the surface.  Also at night, the small true back-radiation effect is not off-set by the absorption of incoming solar radiation by the same infra-red active molecules in the atmosphere.  There is a very important moderation of the surface temperature between day and night due to these effects.  This note was added on 13 November 2014 to remind people of the context in which my criticism of the usual greenhouse gas theory is framed.]

Revised State Real GDP Changes in 2013

According to the U.S. Bureau of Economic Analysis, the percentage change of real GDP in 2013 was:

Now recall that real GDP is under-corrected for inflation due to the government insistence in leaving such volatile, but clearly more rapidly increasing cost, items such as food and energy are not included in the cost of living index.  Then also recall that one should really be looking at real per capita GDP since that tells us whether our standard of living is actually increasing.  If the GDP increases by less than the rate of growth of the population, then our standard of living actually falls.  The average US population growth rate was 0.9% from 2001 to 2010, so real GDP increases of 0.9% provides a stagnant standard of living.

New York state which is running ads nationally claiming to be the second best job creator state in the nation clearly underperformed in 2013 with a state GDP that grew by a mere 0.7%.  The state of Maryland, whose Governor O'Malley believes himself ready to run for the presidency, had a 0.0% growth rate.  Yes, preventing the growth of one's own state's GDP now qualifies a politician well for the Democrat Party nomination for the presidency!

Most of the high growth states in the US are in the center of the nation.  The Dakotas, the Rocky Mountain states, Nebraska, Oklahoma, Texas, and West Virginia are the heroic states for those who wish to earn a living.

Maintaining Climate Change Alarmism: Replacing Cool Weather Station Data with Warm Station Data

Steven Goddard at Real Science has lately had a series of posts on the effects of a very large reduction of temperature data reported from many weather stations since 1990.  The effect of the data loss and its replacement with massaged data, turns out to be a considerable artificial warming effect.  His most recent post is here.

The following plot by Steven Goddard of the older and generally complete temperature data in the USHCN temperature record kept by National Oceanic and Atmospheric Administration (NOAA) until 1990 shows the separated data from the stations still reporting complete data and compares that to the temperature data, then complete, of the stations which are missing much of their temperature data since 1990:

Clearly the stations reporting complete data since 1990 are the stations that were warmer prior to 1990 than those which have been reporting incomplete data since 1990.  The temperature differences between the two data sets prior to 1990 are very substantial.

In an earlier post, Goddard presented the percentage of the NOAA USHCN temperature data for which there was no raw data in the record by year:

Note the increase in the missing raw temperature data since 1990, which became a particularly rapid increase in lost raw temperatures after 1994.  The rate of loss of raw data is really remarkable in the recent years of stagnant temperature reports, indicating that there is a greater and greater need by the alarmists to hide the cooling that must really be going on of late.

Steven Goddard then presents the data as reported since 1990 from these same two sets of stations on the basis of the raw data that actually was reported in the raw data record.

So it becomes very clear that NOAA is preferentially losing the raw temperature data since 1990 from the cooler stations.  The final massaged data that they report to everyone replaces the missing cooler temperatures with temperatures interpolated geographically from the warmer data from the stations with complete raw temperature data records.  That is, NOAA is not just losing cooler raw temperatures, but it is adding in warmer temperatures as though that comes from the cooler stations!

It is clear that many of the NOAA warm stations are warm because of urban and suburban locations, airport locations, and massive heat absorbing or even heat-emitting objects at the immediate site.  The Urban Heat Island Effect has corrupted much of the NOAA weather station network data.  Far from correcting the data from overly warm sites downward, NOAA is correcting the cooler and usually better station data upward.  See Anthony Watts' recent evaluation of the USHCN weather stations.

Over and over, I am asked scornfully by believers in the catastrophic man-made global warming hypothesis if I believe that the government and many climate alarmist scientists are lying to us.  There are many temperature records that have clearly been adjusted with great bias of heart to exaggerate or manufacture a warming trend to support this hypothesis.  I have discussed this in many prior posts.  This very interesting sleuthing by Steven Goddard into the NOAA USHCN temperature data and what NOAA has villainously and corruptly done to the temperature record makes it very clear that our government and its scientists are very determinedly lying to us.  Apparently, the lust for power is overwhelming for many government scientists and for many scientists funded by the power lusting government.

Our Mind-Closing Universities and Conformity

A short post by William Murchison entitled When Higher Ed Shuts the Door on Taxpayer's Right to Know discusses the increasing trend to secrecy on the part of university boards and presidential selection committees.  Murchison says:

The Columbia Journalism Review scores the “increasingly closed-door culture” of university boards, such as Kent State University’s, which not only conducted a secret presidential search recently but “admitted to shredding documents to cover their tracks.”  How about the boards of Wayne State University and the University of Michigan, which not only kept their searches secret but suppressed the names of the finalists until both were hired. Then they told the world!

One can see that given the state of commitment to Progressive Elitism and to political correctness on college campuses that it must be very nearly impossible to conduct the managerial affairs of a public college in the open.  If the board and selection committees maintain political correctness and promote Progressive Elitism as they are expected to by student activists and most faculty members, many taxpayers will be appalled to see what is going on.  If the board and other college controlling committees do not do as expected by student activists and faculty, they are also in very hot water.  Nay, boiling water surrounded by drooling cannibals on most campuses.  Given the culture of the college campus today, such public institutions of Progressive Elitist indoctrination are in a dilemma in all but the most socialistic of states.

Private universities generally only need to satisfy their alumni, students, and faculty, despite raking in huge sums of research grants from the taxpayers.  This is a less tough nut to crack, especially in the Ivy League and equivalent schools, where the alumni have long been very successfully indoctrinated in the Progressive Elitist viewpoint.  These Ivy League schools are very assured that virtually all of their graduates are fully committed to Progressive Elitism, partly because they select for that in their admissions and partly because the on-campus commitment to this indoctrination has long been so thorough.

So how did I escape becoming a Progressive Elitist at Brown University?  By never tiring of pitched debates with up to a dozen Progressive Elitists at a time.  By never being intimated, because I was able to shake the very foundations of their belief, at least for a few hours.  After which they would return as though the discussion never occurred, but with no better ideas than those ideas that had failed so miserably but hours earlier.  Progressive Elitism is definitely a religion for almost all such persons and they refuse to allow that religion to be evaluated and assessed by reason.  But being fully committed to reason, I at least was not susceptible to their religious conformity.

Just as most people in many communities become captured by their desire to belong to a particular dominant religion, whether Christian, Hindi, Islam, or Buddhism, so too do most people in a community dominated by the Progressive Elitist religion just have to be Progressive Elitists so they will belong.  So while people are complex and highly differentiated individuals, most still have a great desire to conform so they will be accepted into a community of people.  If this means denying reality, including their own natures, well then reality will generally be denied.  The human suffering that results knows few bounds, which is why our culture has become more and more cheerless and more and more nihilistic.

Only a community committed to reality, reason, productivity, and individuality can achieve man's greatest needs of security, freely given and valued friendships, real cooperation in mutual endeavors and relationships, the freedom to pursue our highly differentiated values generally according to our individual natures, and hence the freedom to pursue our own happiness.  Such a community is rich in its choices and possibilities.

Unfortunately, those people with already low self-esteem are fearful of that very richness of choices and possibilities.  The less free a society becomes, the larger the number of people with such low self-esteem and the more fearful the society becomes of individuality, choices, and possibilities.  Universities, far from opening young people's eyes to the joy of pursuing life in such a rich, reality-driven society, have long been indoctrinating them to choose a conforming, low-choice, hierarchically managed society that denies them their very individuality.

Is New York State Really a Great Job Creator?

New York state government is spending its taxpayer's money advertising 10 areas near state universities into which a new company or an out-of-state company can move and pay no taxes for 10 years, provided the government likes your business and is given significant controls over your business.  Mostly New York's Democrat Socialist Party controlled government likes certain high technology companies, such as biotechnology companies.

They do not like the vast majority of businesses, at least not enough to give them any tax breaks.  Indeed, most businesses have to be taxed heavily so that the few anointed companies can be lured into the high tax trap of New York and so such advertising can be carried out in Maryland and Oklahoma and many another state.  Even more expensive is the huge welfare state apparatus of New York state.  The only way to fund that redistribution of income is to tax both businesses and better paid persons heavily.

Among the claims in the ad, New York state government claims it has created 400,000 jobs.  They neglect to say over what time period and whether these are net jobs or these are all new jobs with any lost jobs ignored.  The high tax and heavily regulated state of New York is undoubtedly very good at killing jobs, so it is perilous to ignore the jobs destroyed.  According to the Bureau of Labor Statistics, the non-farm seasonally adjusted establishment job tally of private sector jobs increased in New York state from March 2013 to March 2014 by 103,500.  So maybe the 400,000 jobs were net jobs since the depth of the never-ending Great Socialist Recession.  They were not recently created net jobs.  I have ignored any added government jobs since such jobs would only create greater burdens for any private sector company moving to New York.

New York claimed that the 400,000 jobs were the second largest number of jobs created in any state in the US.  In 2010, New York state was the third most populous state, so just to stay even on jobs, it would likely have to produce at least the third most jobs.  Actually, New York state has a population growth estimated from 1 April 2010 to 1 July 2013 of 1.4%, which lags the population growth of 2.4% of the nation as a whole.  What is more, a larger fraction of its population is over 65 years old and a smaller fraction is under 18 years old.  There seems to be no rush of people into New York state owing to good jobs, despite the huge sums of bailout money that the federal government has dumped into New York City and its financial institutions.  That bailout in the form of quantitative easing continues to this day.

Let us examine the growth in the number of non-farm private sector jobs relative to the other states:

There were 26 states in which a higher percentage private sector job increase occurred compared to the 1.45% increase in New York.  This is not such a remarkable achievement by New York as it is represented to be in its ubiquitous ad.  The New New York is not so very different.  It is North Dakota that is the jobs creation giant, albeit in percentage terms only.  Six states have had private sector job growth rates more than twice that of New York state.  In fact, we can see that in this recent period, the claim that New York created the second most jobs in number is clearly wrong.  The population of the state of Texas is greater than that of New York and its job increase of 2.99% clearly means it added many more jobs than did New York.  The population of Florida is about the same as that of New York, so its job increase of 3.49% also means more jobs were created in Florida than in New York.  So, in this time period, at least two states added more jobs than New York did.

The New York state government is running an ad which is very misleading.  There may be some convoluted manner in which it is not a literal lie, but it is surely a serious attempt to mislead the People.

The employment increases in the table above allow us to make a few interesting comments about some other states and the possible presidential prospects for their governors.  Note that Gov. Rick Perry's state of Texas is still tearing up the racetrack in creating jobs with a 2.99% increase between March 2013 and March 2014.  On the other hand, Gov. Chris Christy of New Jersey heads a state government where the private sector job increase is essentially zero at 0.07%, the next to the worst record of any state.  Gov. Scott Walker of Wisconsin has a state with a 1.24% increase, a 32nd ranking counting DC, but at least greater than the population growth in that time.  Gov. Bobby Jindal of Louisiana has a state that is not doing great in the business of job creation either at 1.08% increase and a rank of 35.  On the Democrat Socialist Party side, Gov. Martin O'Malley of Maryland earns his moniker of O'Folley with 40 straight tax increases, a job increase of only 0.52%, and a ranking of 45.  Maryland's population since April 2010 is estimated to have grown by 0.90% a year, so Maryland is not keeping up with the population growth with its meager 0.52% increase in jobs.  I marvel at the ability of some governors to seek higher office when they have not demonstrated that their states can create private sector jobs.

The People should be paying attention to this.  I will acknowledge that some of these states were basket cases and it will take several good governors to allow private industry to turn the states around.  Their state governments have bollixed up the state economies for a very long time in many cases.  The damage done can take a long time to correct and the healing process can take time.  Yet, when good government policies are established, some great results often occur quickly.  The engine of growth is in the private sector, but governments have to stop stuffing the gas tank with sand and sugar.

The Obama Jobs "Recovery"

Let us update the extent of the Obama jobs "recovery."  The most important datum informing us of the health of the economy to provide Americans a means to earn a living is the employment to population ratio.  When many people work, the economy is more likely to hum along fine and operate to raise our standard of living.  When fewer people work, there are more demands for high government extractions of wealth from the productive private sector, the standard of living stagnates or worse, and the people become more dependent upon the Big Government parasite.  So here is the non-seasonally adjusted data on the employment to population percentage according to the Bureau of Labor Statistics:

The fraction of the American non-institutionalized population over 16 which is employed has risen somewhat through April 2014 from a downward drop in the latter half of 2013.  However, this all-critical employment to population ratio is still well below the already very low ratio when he first took the oath of office of the presidency in January 2009.  More critically for Americans, the usual pattern of a recovery from recession within a couple of years to return to a similar employment to population ratio as that prior to the recession is in no way evident.  It is far more accurate to call this employment stagnation in the depths of a never-ending depression than to call this a recovery.

Yet many are shouting harrahs because the so-called unemployment rate dropped in April, even as 800,000 people left the workforce.  Many Americans are in the depth of despair at finding a decent job.  The reports are that most of the jobs that were created in April were relatively low-paying and unskilled jobs.  This is hardly surprising to me given the assertively anti-private sector policies of the Obama Oligarchy Gang of his dictatorial, individual rights trampling, Constitution ignoring, anti-energy, anti-business, micromanaging, lawless, and self-ownership denying regime.  In such a chaotic environment, the private sector does not invest in new business ventures and does not hire.  Obama's claims that money spent by government, which it removes from the private sector, is an investment has been tested many times throughout history.  That experiment always fails to back the statist theory and has failed once again under the Obama Regime.

This remains the never-ending Great Socialist Recession.  Although, given its duration and the degree of human misery it has caused, it is becoming more and more nearly correct to upgrade this condition to the Great Socialist Depression of the 21st Century.  The low employment, the huge transfer of wealth from the private to the government sector, the lawlessness of the government, and the rate of implementation of new socialist programs is very akin to the Great Depression of the 20th Century.  This is what you get when you put government in the hands of the Progressive Elitists.  For all their pretense to know better than most Americans how their lives should by micro-managed by this Progressive Elitist clique, they always manage to mess our society up royally.

We Americans do much better when we insist on controlling our own lives and living them in accordance with our self-chosen values.  Our standard of living is highly dependent upon our rejection of the Progressive Elitist claim that we did not make that.  It is each of us as an individual who must make our own lives and must honor and protect the right of every other individual to do the same.  We must fight off the usurpation of power over our lives by a power-lusting Progressive Elitist cabal.  That cabal refuses to recognize our highly individualistic natures, rich in complexity and highly differentiated.  It refuses to recognize our self-ownership as ObamaCare especially makes clear.  It refuses to recognize our ability to use reason to improve our own lives while exercising our individual rights to life, liberty, property, and the pursuit of our own happiness.  As the Declaration of Independence asserts the protection of such broad individual rights as the sole legitimate justification for a highly limited government and thereby defines the American Principle, the Progressive Elitist movement is profoundly anti-American.

Good Ted Cruz Comment on ObamaCare

From Senator Ted Cruz of Texas:

This is a take-off on the original broken window fallacy explanation given by the economist Frederic Bastiat (1801 - 1850).  Bastiat explained that wealth is destroyed and our standard of living is lowered whenever government promotes destruction in order to provide jobs in reconstruction.  This is a lesson high tax and government spending people such as the Keynesians generally have not learned.  High taxes and government spending are destructive of the wealth that would otherwise reside in the private sector and are almost universally never more than a partial replacement.  In fact, much of the money spent by government has only negative effects on wealth and the standard of living, because the money is only used to hurt the vast majority of the people.  While a few programs may replace 50% of what was destroyed, one must never forget the 50% that remains lost.

Of course the same explanation applies to natural disasters, which was a lesson unlearned by the many who claimed that at least Hurricane Sandy provided some jobs.  A job can easily be provided by destroying net wealth and by lowering the standard of living.  Indeed, we could all easily return to being farmers or hunter-gathers and be most assuredly busy all of the time with the effort to survive.

AGW Theory: Back Radiation Insignificant for Surface Temperature

Introduction

I will examine several energy budgets for the Earth of recent years and address the issue of the extent of radiation absorbed by the atmosphere being re-radiated as long-wave infra-red radiation to the Earth's surface and absorbed there.   This is a very important issue in understanding the role of energy transfer in the lower atmosphere, the troposphere.  This role bears greatly on our understanding of how infra-red absorbing and emitting gases affect the equilibrium climate.  The extent to which increasing concentrations of carbon dioxide in the atmosphere might cause global warming can be estimated with this understanding.  The warming of the surface due to increased carbon dioxide concentrations in the atmosphere is extremely dependent upon the extent of the so-called back-radiation.  If there is little back-radiation compared to the very large back-radiation claimed in the standard model for measuring the surface warming produced by increased carbon dioxide, then there will be much less warming than the minor warming predicted with the standard model.  I will demonstrate reasons to be unsettled about the settled science claimed for the catastrophic man-made global warming hypothesis.

I will explain why these Energy Budgets are not really energy budgets at all.  That is, they do not conserve energy because they do not follow a particular packet of solar insolation absorbed by either the atmosphere or the surface in time and location.  They do not show where that energy is deposited at a particular instant of time along a timeline.  Neither do they follow the flow of that particular packet of energy through the Earth system.  They in fact mix in various energy packets arriving at other times in unquantified ways.  Absorbed back-radiation in the large amounts often claimed in Greenhouse Gas Theory is one clear violator of the need to observe one particular packet of energy at one instant of time or to follow it through the Earth system.  One can use the standard power density approach only if one knows how much infra-red radiation is absorbed by the surface and atmosphere and how much is reflected from each.  Power density is a flow of energy per unit time per unit area, so it is still critical that energy be conserved in the Earth system in addition to maintaining a balance of power fluxes across boundaries.

Still another problem is that the power densities in various forms of energy vary greatly by altitude in the troposphere. Infra-red absorbing gases may change the distribution of energy in the Earth system, but they cannot change the amount of energy absorbed in the Earth system unless they change the reflectivity of the atmosphere or the surface.  The condensation of water vapor is an obvious means of changing the reflectivity of the atmosphere, though it is hard to quantify the resultant effects.  A less obvious change of reflectivity might be due to increased carbon dioxide encouraging so much more plant growth that land mass reflectivity changes.

If we focus on the surface temperature as particularly important to mankind and our quality of life, we should be very interested in the manner in which energy is transported away from the surface by the atmosphere.  We have to note that the lower troposphere plays the most direct role in our weather.  The surface is heated by solar insolation transmitted by the atmosphere and absorbed by the surface.  The cooling effects on the surface are dependent on the emissivity of the surface, the loss of energy due to water evaporation, and direct conduction due to air molecules colliding with the surface.  Water evaporation soaks up large amounts of heat and as water vapor is carried aloft, it condenses into liquid or ice commonly at altitudes of about 2 to 4 km, at which point a great deal of energy is released to heat the atmosphere.

Some of the heat energy that is radiated as long-wave infra-red radiation from the surface is transmitted through the atmospheric window directly into space.  Some is absorbed by water vapor or by carbon dioxide in the atmosphere.  Because of the high collision rates of molecules in the lower atmosphere, the absorbing water vapor or carbon dioxide molecules will most often lose the absorbed energy to surrounding nitrogen and oxygen molecules or to argon, before they can re-radiate the absorbed infra-red radiation that originated from the surface.  This is one way in which the rate of loss of heat energy decreases in the lowest part of the troposphere with increasing altitude very close to the surface.

The amount of radiant energy transfer then stabilizes in the first few hundred meters altitude at a level dependent upon the density of water vapor and other infra-red absorbing and emitting gases such as carbon dioxide.  The long-wave infra-red emitted from these infra-red active gases is radiated at the speed of light until it is absorbed again.  That energy transport is faster than the convection or the evaporation/condensation mechanisms of water.  This faster radiation transport of energy to higher altitudes is a cooling effect, so higher concentrations of infra-red absorbing gases have a cooling effect in a negative feedback or a counteracting relationship with any warming caused by back-radiation.

Greenhouse Gas Theory says that the long-wave radiation from each molecule is emitted isotropically in all directions.  All of the downward radiated long-wave infra-red is absorbed by the surface, which is said to be a black body absorber for such long-wave radiation.  I will show that Energy Conservation assures us that the surface does not absorb down-welling atmospheric radiation, but either reflects it in its entirety or that more likely the supposition of isotropic molecular emission is actually incorrect because infra-red photons are created by a directional field dependent upon the temperature gradient.  The upwelling radiation has a magnitude determined by the temperature gradient over the mean free path length of the radiation at a given wavelength.  Both the upwelling and the down-welling infra-red radiation are less than is normally depicted in the standard model of catastrophic man-made global warming. 

Because no down-welling long-wave infra-red radiation is absorbed by the surface in the usual equilibrium condition of a cooler atmosphere over a warmer surface, the upwelling long-wave radiation is the only radiant energy transport mechanism. Thus the emission of infra-red radiation by water vapor or carbon dioxide is a very rapid mechanism for the cooling of the lower troposphere.  That mechanism becomes less important as the concentration of the absorbing/emitting molecules decreases with altitude relative to nitrogen and oxygen, which causes the emissivity of the atmosphere to decrease with altitude.  Convection then again becomes a more important mechanism for energy transport to higher altitudes, but this is a slow energy transport mechanism.  Water vapor density decreases more rapidly than does the overall atmospheric density.  Since water vapor is much the dominant infra-red active gas, its role in rapid energy transport rarely extends to altitudes greater than 4 km.

The more infra-red radiating molecules in the atmosphere, the more energy is rapidly transported upward through the atmosphere.  Air humidity can vary greatly and it has a major impact on the rate of temperature change with altitude.  The more water vapor in the air, the smaller the temperature gradient with increasing altitude.  This is due to the increase in rapid transport of energy upward due to the increased emissivity of the air when it is humid.  Higher humidity causes a powerful cooling effect because it is maintained by increased surface evaporation and causes increased radiative losses as the air infra-red emissivity increases with added water vapor.

If there were increased down-welling radiation in accordance with Greenhouse Gas Theory and it were absorbed by the surface when the humidity was high, then the surface temperature would be greater when the humidity was high.  The data clearly shows that the surface is cooler when the humidity is high, which clearly shows that the surface is not absorbing added down-welling long-wave infra-red radiation.  Consequently, the increased emissivity of air due to high humidity causes a decrease in surface temperature because the direction of the emitted long-wave infra-red energy is in the upward direction.  Increased carbon dioxide will also cool the lower troposphere, though it increases air emissivity about 2.5 times less than does water vapor for a given partial pressure and its concentrations are much lower.

Both water vapor and carbon dioxide also absorb some of the incoming solar insolation and so have a surface cooling effect by decreasing the atmospheric transmission of solar energy.  The evidence is mounting that these infra-red absorbing gases do more to cool the lower troposphere than they do to warm it.  The warming effects are due to the rarer condition when the lower troposphere temperature is greater than the surface temperature and due to the narrowing of the atmospheric window through which radiation from the surface is not absorbed anywhere in the atmosphere.  These warming effects prove to be smaller than the cooling effects are, contrary to the Greenhouse Gas Theory which wrongly proclaims the narrowing of the atmospheric window by infra-red active gases to be the only important effect on surface temperatures.


Earth Energy Budgets

Let us discuss the amount of power density in the atmosphere which is absorbed by the surface due to surface-incident infra-red radiation from the atmosphere since we see from the above introduction that this is a critical issue for surface temperatures.  This paper will calculate the equilibrium contribution for such absorbed power from the atmosphere and therefore of energy which can become added heat in the surface.  Some comments on limits for non-equilibrium absorption of radiated energy from the atmosphere will also be made.

Let us consider a few versions of the Earth's energy budget first and then I will discuss the implications for the adsorption of energy at the Earth's surface due to down-welling atmospheric infra-red radiation.  An energy budget currently posted by NASA is shown in Figure 1 below.




Figure 1. The principal NASA energy budget for the Earth as of February 2013. Note the huge surface radiation and the huge radiation from the atmosphere all of which is absorbed by the surface. The surface-absorbed atmospheric down radiation is 100% of the solar insolation at the top of the atmosphere and it is all claimed to be absorbed by the surface. The greenhouse gases absorb the radiation emitted from the surface at a 105% level and they also absorb 23% of solar insolation for a total of 128%.  They emit radiation at a power density of 159%.  But only 71% of solar insolation in total was absorbed by the entire Earth system!


It is interesting to compare this recent energy budget with the energy budget briefly found earlier on a NASA website and still found in an educational resource here. This energy budget looks like this:





Fig. 2. This ephemeral NASA energy budget showed a much reduced radiation emission from the surface of only 21% and no radiation from the atmosphere to the surface.  The IR-active gases radiate a power density of 64%, compared to the 159% radiated in Fig. 1.  The entire Earth system absorbed 70% of solar insolation and no energy transport mechanism claims to use more than 70% of the power.

We will compare these more recent energy budgets from NASA with the Kiehl-Trenberth energy budget of 1997 which was featured in the UN IPCC 4th assessment report of 2007. That budget is shown below with my conversions to percentage of the solar insolation power density at the top of the atmosphere.




Figure 3. The Kiehl-Trenberth energy budget for the Earth of 1997, which was featured in the UN IPCC report of 2007. This energy budget claims a 114.0% emission of radiation from the surface with a power density of 89.7% absorbed by IR-active gases in the atmosphere. Radiation from the atmosphere to the surface is 94.7% and it is claimed to be entirely absorbed by the surface.  The infra-red active gases emit a total infra-red radiation of 151.7% when only 68.7% of energy was absorbed from solar insolation by the entire Earth system.  


The items in the several energy budgets can now be compared. I will try to determine the best likely values for the various energy items while treating any down radiation from the atmosphere which is absorbed by the Earth's surface as a variable.  The percentages are referenced as they are in the energy budgets of Figs. 1 - 3 to the average solar insolation at the top of the atmosphere.  In the apparent manner of the Energy Budgets shown above, I will partition the energy absorbed by the surface from solar insolation among Conduction/Convection, Evaporation of Water, and Infra-red Radiation as the energy outflow mechanisms of the surface.  But, it should be remembered that even a few hundred meters above the surface, much of the surface infra-red radiation has been absorbed by water vapor and carbon dioxide or other infra-red active gases.  Those IR-active gases transfer much of the energy they have gained by absorbing energy radiated from the surface through many collisions to the far more numerous non-absorbing air molecules.  This is a conversion of infra-red radiation energy transport into conduction/convection energy transport.  So, the radiant power and the air conduction/convection energies are not constants in the lowest part of the troposphere for this reason.  Then the density of infra-red absorbing and emitting molecules continue to decrease and the radiant energy component falls off further.

I will discuss these energy item by energy item.  After discussing the energy items, I will present a number of mathematical relationships they must maintain for energy conservation purposes.  The abbreviations for each will then be used in the expressions of these relationships.

Atmospheric Reflection (ARfl): This is reflection from the boundary layers of the atmosphere, from aerosols, and from clouds. The energy budgets vary from 22.5% to 26%. But what is more nearly constant is the sum of the atomspheric and the surface reflectivities at 30%, 30%, and 31.3%. The greater problem is in how the reflectivity is divided by type. The most recent NASA values ought to be the best, especially given that there is no real advantage to the global warming alarmists as long as the sum of the surface reflection and the atmospheric reflection is 30%. The newest values also lie between the older values.

Surface Reflection (SRfl): See the discussion under Atmospheric Reflection. In addition, a 4% reflection as stated in the NASA 2011 budget is clearly too low a reflection value in my experience with UV, visible, and shortwave IR on surfaces.

Atmospheric Absorption (AA): This is a hard value to measure by itself directly, but it is the difference between 100% and the sum of the Atmospheric and Surface Reflections (30%) and the Solar Surface Absorption.

Solar Surface Absorption (SSA): I am weighing the NASA 2011 energy budget more heavily on this one since that actually opens the door more widely for some downward radiation absorption from the atmosphere by the surface.  This will be more apparent later in the discussion.

Conduction, Convection from Surface at the Surface (CCS): I simply averaged the three Energy Budget values.

Evaporation (E): I just averaged the three energy budget values.

Surface Radiation at the Surface (SR):This has two components, namely that fraction of the solar insolation absorbed by the surface which is re-emitted as infra-red radiation and any energy re-emitted as infra-red which was absorbed from down-welling radiation from the atmosphere.

Surface Radiation Emitted to Space without absorption by the atmosphere since the radiation is in the atmospheric window (SRW): I set this in accordance with the most recent NASA data since this should be a very measurable quantity.  The atmospheric window has surely been well-understood due to its many military applications in sensor technology.  Setting this value higher actually makes the Surface Radiation at the Surface (SR) higher, which might allow the absorbed down-welling radiation value to be greater.

Surface Radiation Absorbed by Atmosphere (SRA): The sum of this and the Surface Radiation Emitted to Space in the Atmospheric Window (SRW) must equal the Surface Radiation immediately at the surface. This value was fixed at the largest value it can have.  The basis for that will be explained later.  This is an important issue.

Atmospheric Radiation Absorbed by the Surface (ARAS): This is any down-welling radiation from the atmosphere, including back-radiation, which is actually absorbed by the surface.  It is assumed here that it is possible for some of this radiation to be reflected from the surface without absorption.  That reflected energy may appear to have been emitted from the surface.

Total Radiation Absorbed by Earth and Emitted to Space (TRAS): This value clearly should be 100% minus the reflected sum, which all accounts set at or near 30%.

Atmospheric Radiation Emitted into Space (ARES): This plus the Surface Radiation Emitted to Space through the Atmospheric Window (SRW) must equal 100% minus the solar insolation reflected back to space, which is about 30%.

% of Surface Radiation Absorbed by Atmosphere: This is just the (Surface Radiation - Surface Radiation Emitted to Space in Window)/ (Surface Radiation).

The following relationships must be maintained in the Earth Energy Budget in order to be consistent with the Conservation of Energy, though their doing so is not a guarantee of Energy Conservation, as we shall see:

ARfl + SRfl + AA + SSA = 100%

CCS + E + SR - ARAS = SSA

SRW + SRA = SR

TRAS = 100% -  ARfl - SRfl

TRAS = ARES + SRW

These conditions are satisfied by my Earth Energy Budget.


How Much Down-Welling Radiation is Absorbed by the Surface?

Let us consider the temperature profile with respect to altitude for the Earth's atmosphere to perform the calculation of the maximal radiation absorption possible by the atmosphere of any infra-red radiation emitted from the Earth's surface.  That profile is shown in Fig. 4. below:

Fig. 4.  The temperature and density profiles of the Earth's atmosphere are shown as a function of altitude in this plot.  Initially as the altitude is increased, the temperature falls in a linear fashion and then becomes constant for a distance.  The linear decrease is in the troposphere and the constant value is in the tropopause.  The constant value is about 216.65 K.  The temperature never becomes as cold again until above 70 Km altitude, where the atmospheric density is virtually zero.  There are no significant numbers of greenhouse gas molecules at this altitude.


With the Earth's surface at 287.65K and the coolest part of the atmosphere with infra-red absorbing gases in it at any reasonable density being at the top of the troposphere and in the tropopause, the maximum emission of power density by radiation from the surface, assuming no other power losses due to other surface cooling mechanisms, is given between the surface and the top of the troposphere by:

P = ε_s σ T_s⁴ – ε_t σ T_t⁴,

where T_s is the surface temperature and T_t is the temperature at the top of the troposphere and in the tropopause.  For now, I will assume an emissivity of one here.  In reality, the emissivities are not well-known, but are less than one.  The first term is the larger term, so reducing the emissivity of that term does the most to decrease the maximum power generated by the surface due to its temperature.  One can make P larger by making the top of the troposphere emissivity smaller also, but that comes at a big cost in that it means opening up the atmospheric window and losing the possibility of the atmosphere absorbing much of this power density.

Consequently, the maximum emitted power density from the surface under the black body assumption is

P = σ (287.65K)⁴ – σ (216.65K)⁴ = 263.26 W/m²

Comparing this to the average solar insolation at the top of the atmosphere, we find that the percentage of the maximal radiation from the surface which can be absorbed by the atmosphere is

100 (263.26 W/m² / 341 W/m²) = 77.2%.

Now we have the maximum radiation the atmosphere can absorb from the surface due to its temperature being 287.65K and if we add the 23% of the solar insolation absorbed by the atmosphere according to the February 2013 NASA Energy Budget of Fig. 1., we find that the atmosphere cannot have more than 100.2% of the energy absorbed even under the unphysical assumptions of this Energy Budget.  Yet that 2013 NASA Energy Budget claims the atmosphere radiates 100% power density to the surface at the same instant it is radiating 59% to space, for a total of 159% radiated power density from the atmosphere. Not only does this so-called Energy Budget fail to conserve energy in reality, but it also fails to conserve the energy sums in the fantasy world it has created.

Energy Conservation immediately tells us that the maximal absorbed power by the atmosphere from the surface radiation value calculated above is too high, however.  Recall that 30% of solar insolation power was lost due to reflection, so this power was never absorbed by the Earth system.  The equilibrium earth is losing power into space carried by infra-red radiation to the tune of 70% of the incoming solar insolation.  The total amount of power density in the Earth system of atmosphere and surface is 70%.  The black body calculation of the power absorbing capacity of the atmosphere might have proved to be lower than 70%, but it is higher, so it does not provide the upper limit.

What is the maximum power that the atmosphere can radiate anywhere?  In my budget, it absorbs a power density of 20% of solar insolation from the incoming solar radiation, AA.  Of the 50% of solar insolation absorbed by the surface, 12% = SRW was sent through the atmospheric window into space.  The other 38% that enters the surface is transferred to the atmosphere.  The atmosphere had absorbed 20% of its energy directly from incoming solar insolation, AA.  Thus, the most power density in the atmosphere, ARES, is AA + SSA - SRW = 20% + 50% - 12% = 58%.  In the equilibrium condition, energy flows into the atmosphere at a rate of 58%.  

Now the entire Earth system absorbed energy at a rate of 70% (TRAS).  Of that, the surface emitted 12% directly into space without absorption by the atmosphere.  Subtracting 12% from the total Earth system power density outflow of 70% at equilibrium, we find that the atmosphere has to supply an outflow of radiation into space, ARES, at a rate of 58%.  But, 58% power density or energy flow rate density is all the atmosphere has to supply, because that is the power inflow rate.  Therefore, the atmosphere is not supplying the surface with any power density in this equilibrium condition.  The most it can supply any power sink is 58% and that is going into space, not the Earth's surface.

This energy conservation requirement is seen in the schematic diagram below:

Fig. 5.  The absorbed energy from the solar insolation is 70%, becoming 20% absorbed by the atmosphere and 50% absorbed by the Earth's surface as shown on the left.  The 30% of the solar insolation reflected does not affect temperatures in the Earth system as long as the absorbed energy remains constant.  The surface gives up a power density of 12% straight to space and 38% is absorbed by the atmosphere.  The atmosphere now has 58% of energy, which it is radiating off into space.  The surface never sees more than half of the photon energy entering the Earth system, while the atmosphere never sees more than 58% of the incident solar insolation energy at the top of the atmosphere.


This approach to the energy flow is actually following the path taken by portions of the energy of a particular packet of solar insolation energy that enters and leaves this equilibrium Earth system.  What we find is that ARAS must equal zero in equilibrium because the atmosphere does not have the energy flow both to supply ARAS to the surface and to supply what it must to the space power sink.

Let us review:

On the input side:  ARES = TRAS - SRW = 70% - 12% = 58%

On the output side:  ARES = AA + (SSA - SRW) = 20% + (50% - 12%) = 58%

Since ARES is the entire supply of energy flow in the atmosphere and 58% is emitted as infra-red radiation to space, there is no additional energy flow to provide a flow of energy to the surface and to be absorbed by it.  Thus, energy conservation tells us that ARAS = 0%.

There are many additional ways using energy conservation that can easily show that the upper bound on ARAS is many times less than those of the Energy Budgets of Fig. 1 and Fig. 2.  Even in non-equilibrium, but still annual period conditions there are obvious limits on the amount of power density that the atmosphere can send as down-welling radiation which is absorbed by the surface.  Given that the atmosphere has at most 58% of solar insolation at the top of the atmosphere, it cannot possibly radiate more than 58% power density toward the surface.  Indeed, if it did that, the temperature of the atmosphere would be 0 K and it would be absurd to believe that emitters approaching 0 K could emit any but negligible power density.  If the Greenhouse Gas Theory assumption that the atmosphere radiates energy both up and down is made, it would be impossible that it could radiate down even half of this power density, that is 29%.  Since Earth-wide the transport of energy upward in the lower troposphere will never cease due to a total loss of convection and water evaporation, the transient annual limit on down-welling radiation absorbed by the surface must be much lower than 29%, indeed lower than 14.5%.  These are just some of the obvious inconsistencies of the NASA and UN IPCC energy budgets with the Law of Conservation of Energy.

The comparison of these other Earth Energy Budgets with my own is then:

Let me note that ARAS, the power density of long-wave radiation from the atmosphere absorbed by the surface is not always zero in transient periods.  Of course there are cases when the air moving over a surface is warmer than the surface is.  These transient cases are not the equilibrium case and the values for SR, SRA, and SRW should all have been averaged over transient effects such that their values are those of the equilibrium condition.


On Radiation from the Earth System into Space

Now the classical and simple radiation calculation that gave plausibility to the claim that infra-red absorbing gases such as water vapor, carbon dioxide, and methane warmed the Earth was based on the following single emission temperature idea:

P = σ T⁴, where P is the total solar power absorbed by the Earth and T is the single infra-red emission temperature of the effective emitter, which is somehow broadly the Earth.  The Earth system is assumed to be a black body emitter.  Using the fact that about 70% of the solar insolation at the top of the atmosphere is absorbed in the Earth system, while 30% is reflected, we get P = 0.7 (341 W/m²), T = 254.73 K.  This is a temperature 33K below the Earth's surface temperature, so many assumed some radiative mechanism had to be found to explain this and their hypothesis was that it was due to greenhouse gases.  I will show a simple calculation below that suggests that the explanation is actually in large part a non-radiative one.

One of the interesting consequences of the Energy Budgets results from the manner in which the radiative power density into space is partitioned between an origin from the surface and from the atmosphere.  The total Earth system radiation into space, P_E, is 

P_E = P_s + P_a ,

where P_s is the power density emission from the surface through the filter of the atmosphere that is emitted into space without any atmospheric absorption and P_a is the atmospheric emission into space. From my Energy Budget, P_E is 70% of solar insolation incident upon the top of the atmosphere, P_s is 12% of solar insolation, and P_a is 58% of solar insolation.  If we give these the form of a sum of Stefan-Boltzmann power density emissions we have

P_E  = ε_s σ T_s⁴ + ε_a σ T_a⁴

P_E = 0.70 (341 W/m²) = 0.12 (341 W/m²) + 0.58 (341 W/m²), so

ε_s σ T_s⁴ = 0.12 (341 W/m²) = 40.92 W/m²

This implies that with T_s = 287.65K,

ε_s = 0.1054,

which is much lower than the surface emissivity which would be measured immediately above the surface when infra-red radiation is the only energy transport mechanism.  This low effective emissivity is not only low due to energy going into evaporation and conduction transport of energy at the surface-atmosphere boundary, but it is also reduced by the effect of the entire troposphere above the surface.  Among those effects are the short mean free path of radiation which can be absorbed and emitted by water vapor and carbon dioxide.  The very low effective emissivity of the surface is that as seen through the filter of the atmosphere from space.  It is a measure of how poor the radiative equilibrium of the surface is with space.  This implies the dominance of other energy transport mechanisms in the lower troposphere than radiation.

We can also calculate the effective atmospheric emission temperature, but we have a poor knowledge of the effective atmospheric emissivity, ε_a, so we will assume it to be 1.0 for now.  So,

σ T_a⁴ = 0.58 (341 W/m²) = 197.78 W/m²

T_a = 243.03 K

This implies that the weighted average temperature from which the Earth's absorbed solar power is emitted back into space is

(0.12 T_s+ 0.58 T_a) / 0.70 = 250.68 K,

which is somewhat lower than the often mentioned 255 K due to making no distinction between the surface and the atmospheric long-wave infra-red emissions and treating the Earth as a black body as was done in the opening paragraph of this section.  It is interesting that the Earth surface temperature can be maintained at 287.65 K and the weighted average source emission temperature of the atmospheric emission is so little changed relative to the simple-minded, but claimed to be critical, single temperature emission source model.

If the atmospheric emissivity is taken to be less than 1, then the effective atmospheric emission temperature increases. The claim is often made that the emissivity of water is about 0.95.  So water being the dominant atmospheric infra-red emitter, let us take the atmospheric emission to be 0.95. Then

 0.95 σ T_a⁴ = 0.58 (341 W/m²)

T_a = 246.16 K,

which agrees pretty well with the brightness temperature of the Earth emission spectrum seen from space at those frequencies for which extensive water vapor absorption of infra-red radiation occurs.

Then the weighted source temperature becomes

(0.12 T_s+ 0.58 T_a) / 0.70 = 253.27 K.

This is very close to the single emission temperature black body model result we calculated in the first paragraph of this section.
The effective atmospheric temperature, T_a, and emissivity, ε_a, needed to give a weighted source temperature of 254.73 K can now be calculated to be

T_a = 247.92 K

ε_a= 0.923.

This effective total atmospheric emission temperature in the U. S. Standard Atmosphere is at an altitude of about 6200 meters and well below the top of the troposphere, but also about 2 Km higher altitude than is affected by the major variations in near surface humidity.  The number of molecules per cubic meter at 6000 meters is 1.298 x 10²⁵.

It is interesting to note that the Earth emission spectrum where CO₂ absorption occurs has a characteristic temperature in the 200 to 210 K range, which is in the upper mesosphere or above 75 Km in altitude in the U. S. Standard Atmosphere.  Its density there is extremely low.  The number of molecules per cubic meter is only 8.300 x 10²⁰, which is only 6.4 x 10^-5 times the molecular number density of the atmosphere at 6 Km near the effective atmospheric emission temperature.  In addition, the emissivity of carbon dioxide is probably less than 0.4 that of water per molecule, so the power density emission of CO₂ into space is extraordinarily lower than that of water.

Does Down-Welling Infra-Red Radiation Reach the Surface in the Equilibrium Condition?

We now know that long-wave down-welling infra-red radiation from the atmosphere is not absorbed by the surface under equilibrium conditions due to our application of the Law of Conservation of Energy.  This could mean that it is simply reflected from the surface.  But, let us now examine whether it can be expected to even reach the surface in the equilibrium condition when the atmosphere has a slowly decreasing temperature gradient with altitude.

One can look up from the surface and detect radiation from infra-red emitting molecules when using an infra-red thermometer or pyrometer.  These instruments always measure the effective temperature of those emitting molecules, so everyone concludes that the pyrometer is measuring the radiation that every emitting molecule emits in all directions as it would if it were the only emitter surrounded by vast domains of space at a 0 K.  But this picture is not the case when examining the surface of the Earth with an envelope of atmosphere with low concentrations of infra-red active molecules for a short distance and then the vast reaches of space at near zero temperature.

In the Earth system, the distributed infra-red molecules have a range of temperatures from very near the surface temperature to temperatures of about 216 K at the top of the troposphere. Instrument measured down-welling long-wave radiation spectra are complex with various frequencies represented in distributions not characteristic of the complete broadband Stefan-Boltzmann black body radiator distribution.  In addition, there are uncertainties about which frequencies will be absorbed in the various materials and in the complex surface of the Earth. Bear in mind that ARAS, which is zero when the Earth system is in equilibrium, is the Atmospheric Radiation Absorbed by the Surface.  That it is zero says nothing about the flux of radiation coming from the atmosphere and measured by a device looking up from the surface.  It only says that it is not absorbed by the surface.

If there is any such down flux of long-wave power, it is entirely reflected from the surface or it does not actually reach the surface, when the atmosphere is cooler than the surface, which in equilibrium conditions it is.  Any such reflected power flux in the atmosphere would be difficult to distinguish from radiation actually emitted from the surface, given that the mean free path for infra-red radiation from water vapor or carbon dioxide in the atmosphere is very short near the surface.  Therefore, that radiation comes from molecules only very slightly cooler than the surface.  But that reflected energy is also going to be re-absorbed by water vapor or carbon dioxide only a short distance above the surface.  This means that the entire process of infra-red emission from one molecule a short distance above the surface, surface reflection, and re-absorption by another molecule a short distance above the surface is a non-event as far as energy transfer is concerned.  This is not true for up-welling long-wave infra-red emissions.  When that infra-red energy is absorbed, half is emitted upward to lower density atmosphere and travels a longer distance before absorption than is the radiation emitted downward.  The density gradient and the reflection of cooler infra-red radiation from the warmer surface results in a net up-welling of radiant energy.  This is the opposite of what Greenhouse Gas Theory maintains, but it is consistent with observational measurements of temperatures.

That reflected infra-red radiation is a factor in measuring the temperature of emitters with infra-red thermometers or pyrometers is noted in this figure from the Fluke 56x Infra-red Thermometer User's Manual:

Fig. 6.  Infra-red thermometers or pyrometers measure both the infra-red radiation emitted from an object and the infra-red radiation it reflects.  The reflected infra-red from shiny objects and from objects at temperatures not much different than their surroundings are especially problematic.  One is often advised to apply a matte black tape to a surface which is shiny and to measure the temperature of the tape in the hope it is not much different than that of the object in order to reduce errors from reflected infra-red radiation.  This figure is from the Fluke 56x User's Manual.

Most pyrometers measure temperatures by measuring the infra-red radiation in the 8 - 14 micron wavelength range.  Those designed to measure temperatures near ambient or cooler than ambient temperatures will have a sensor which is cooled by a thermoelectric cooling device based on the Peltier effect.  Such devices can cool to temperatures about 70K below the ambient temperature, which is why most pyrometers have a lower temperature measurement level of -30 to -40 C.  Use of a cooled sensor lowers the noise level and it establishes a known temperature energy sink for energy radiated from the object whose temperature is to be measured.  The sensor temperature can be directly measured by a built-in thermocouple and then used as a known temperature reference for the remote object.  Because of that cooled sensor, the measuring device is not a passive device.  It alters the electromagnetic field in its vicinity by introducing the lower temperature region of its sensor.  The consequence of this is that in order to measure the emitting object temperature more accurately, it actually changes the flux of infra-red radiation in its vicinity.  A pyrometer does not measure the infra-red radiation that was in that vicinity before the pyrometer perturbed the radiation field. It is designed to measure temperatures, not to measure radiation power densities.

The important property that defines a black body cavity is the fact that its interior has a constant energy density.  The chemical potential of black body radiation is zero, which means that more photons are automatically generated by the cavity walls if the volume of the cavity is increased to maintain the constant energy density inside the cavity.  U/V = u = aT⁴, where U is the total interior cavity energy, V is the cavity volume, and a is Stefan’s constant.  Very close to the exterior surface in vacuum with the black body emitter surrounded by space at 0 K, u = aT_s⁴, where T_s is the surface temperature, just as is the case in the interior of a black body cavity.  The energy of the photons per unit volume is equal to the energy density u.  The energy density of radiation decreases with the square of the distance from this radiator.  The black body radiator radiates energy because its temperature is a measure of the kinetic energy in its material and that kinetic energy is causing electric dipoles to generate an electric field.  That electric field strength diminishes with distance.  The field generates photons that travel along the field lines into the distant lower field strength regions, which are at a lower temperature.

Let us place another black body radiator a relatively short distance h away from the first black body radiator at a temperature T_a.  It perturbs the electric field created by the first black body radiator’s oscillating electric dipoles.  Its own oscillating dipoles would in isolation being emitting radiation isotropically into space, but because of the other black body radiator a short distance h away from it, electric field lines will be developed between the two bodies.  If the temperature T_a is less than that of T_s, the field will lose amplitude as it approaches this second body.  The field will cause photons to travel from the stronger field at the higher temperature body toward the lower field strength at the lower temperature body.  This is just as happens when a single body is present and all the photons follow the diminishing strength field lines into the far distance of space.  The photons flow from the strong field developed by the oscillating electric dipoles to the weak field sinks.  When the body with T_a less than T_s is inserted, all photons still travel from the higher temperature body with the stronger electric field to the lower temperature body with the weaker electric field near it.

If the usual picture of two black bodies radiating photons at the rate characteristic of such bodies when isolated in space and far from anything not at or very near 0 K were true, then when we examine the space on the direct line between these bodies the value of u = a ( T_s⁴ + T_a⁴ ), that is the photon density is the sum of the radiated densities for each isolated body.  But we know that this is not the case.  The energy density is actually u = a ( T_s⁴ - T_a⁴ ).  If the usual assumption of photon flow in both directions were true and each photon stream does not interact with the stream of photons from the other black body emitter, then arbitrarily close to the surface of either black body radiator, the energy density would be greater than it would be inside a black body radiator cavity.  If T_s and T_a were equal, we would have the particularly nonsensical result that the energy density arbitrarily close to the outside surface of either body would be twice that found inside the cavity of such an ideal black body!  Therefore, photons only complete the travel from the warmer body to the cooler body with the power density given by the usual relationship that P = σ (T_s⁴ – T_a⁴).

Consider the surface of the Earth for a moment.  The density of atoms per cubic meter in the surface is about 1 gram per cubic centimeter for the water that covers about 71% of the Earth's surface and even greater for land materials or if considering the salts in the oceans.  Expressed as a density per cubic meter of water, this is 1000 kg per cubic meter.   A cubic meter of water has 3.34 x 10²⁸ molecules.  Infra-red emission and absorption in a surface occurs mostly in the outer 2 micrometers of the surface.  Consequently, there are about 6.7 x 10²² water molecules emitting infra-red at most from the surface.  These are the same surface molecules that absorb any incident infra-red radiation on the water surface.

The density of the atmosphere near the surface at sea level is 1.225 kg per cubic meter in the U.S. Standard Atmosphere at 288.15K.  The number of molecules per cubic meter at sea level in the U.S. Standard Atmosphere is 2.55 x 10²⁵/m³.  The important infra-red active gas near the surface is water vapor and its density per cubic meter is commonly between 10 g/kg to 14 g/kg of air as shown in Fig. 7. below.  At a specific humidity of 12 g/kg, the number of water molecules/m³ of air is about 4.9 x 10²³.  Therefore, there are more water molecules in the first cubic meter of air above 1 m² of water surface emitter or absorber molecules by a factor of 7.3.  This should mean that radiation that can be absorbed by water vapor will be absorbed in the first meter of air above the surface at a humidity near the Earth average humidity.

Fig. 7. The average specific humidity of air is shown as a function of latitude.  The specific humidity is the weight of water in grams in one kilogram of air.

In comparison, at the current 400 ppm of carbon dioxide, there are 1.0 x 10²² molecules of CO₂ /m³.  There are 49 water vapor molecules for every carbon dioxide molecule at a specific humidity of 12 g.kg.  In addition, the emissivity of CO₂ molecules is less than 40% that of water vapor molecules.  Many of the carbon dioxide molecule absorption peaks are largely or mostly already absorbed by overlapping water vapor absorption lines, so the effect of CO₂  near the surface is very minimal compared to that of water vapor.  This is less true at altitudes above 4 km when water vapor is very low, but carbon dioxide maintains a proportional mixing with nitrogen and oxygen.  However, such altitudes have nothing to do with the back-radiation issue.

Returning to the back-radiation caused by water vapor, we find that the temperature difference over the 1 meter absorption range for the dry air of the U.S. Standard Atmosphere is only about 0.0065 K.  For humid air, the temperature gradient is even less.  So if the water surface and the water vapor in the first meter of air above the surface are treated as gray bodies, we have a power transfer from the surface to the water vapor in the atmosphere of P_W

P_W = σ (ε_s T_s⁴ – ε_a T_a⁴) , 

for the surface s and the atmosphere water vapor a and if we take the emissivities to be 0.95, this is equal to 0.033 W/m².

However, water vapor does not absorb infra-red across the entire surface emission spectrum which has a black body radiation distribution with energy.  It actually absorbs only about 65% of it.  So the actual water vapor absorbed infra-red radiation is only about 0.021 W/m².  But because of the 6.9 x 10⁹ collisions/s, most of this energy is transferred to non-radiating nitrogen, oxygen, and argon.  Only about 20% is re-radiated and half of that is radiated toward space.  Consequently, the total back-radiation, P_B, is about

P_B = (0.2) (.5) (0.021 W/m²) = 0.002 W/m²

Thus, the absorbed back-radiation has an upper limit of about 0.0006% of the average solar insolation at the top of the atmosphere (342 W/m² ) !  For all intents and purposes, the absorbed back-radiation is zero.

Conclusion

So, I conclude that the radiation from atmospheric molecular emitters that can be radiated back to the surface is very nearly zero thanks to the short absorption lengths of infra-red that can be absorbed by water vapor and the high molecular collision rates near the surface in the atmosphere.

We exaggerate the energy of down-welling radiation by using active sensors to measure temperatures.  The sensor purposely inserts a cooler region to establish a low-field sink with known temperature to make an accurate measurement of the temperature of bodies it needs to determine.  Many then wrongly infer the measured photon flux was there prior to the measuring instrument being introduced into the many-body electric field.  They assume the photons emitted from an object of the measured ​temperature were previously present.  The instrument is active, not passive as assumed.

The two energy budgets of Figs. 1 and 3 with large back-radiation absorption by the surface posit the deposition of large amounts of power in the atmosphere.  The atmosphere simply does not have enough energy flow into it to send 100% of the solar insolation back to the Earth as infra-red radiation and also send 59% of solar insolation equivalent energy into space, which is the claim of Fig. 1.  There is a problem here with energy conservation despite the power fluxes being in near balance in that NASA 2013 Earth Energy Budget.  The atmosphere simply cannot be flinging 159% of the energy from the sun, especially when 29% of the solar insolation is never absorbed by anything because it is simply reflected and since the surface is flinging a power flux of 12% into space with no interaction with the atmosphere at all.  So, this energy budget would have us believe that 100% - 29% - 12% = 59% of the solar insolation so heats the atmosphere that it can fling out 159% of solar insolation energy equivalent.  This is unphysical and in violation of the Conservation of Energy.

It is also clear that the Fig. 1. NASA Energy Budget violates Conservation of Energy at the surface.  The entire energy flow rate into the Earth system is 71% of solar insolation, yet it claims to have energy flowing out of the surface at a rate of 117% of the solar insolation and flowing into the surface at a rate of 100% for a total energy flow rate of 217% compared to a maximal upper limit of 71%.  Now why has NASA claimed that the energy flow rate out of the surface is 117%?  It is perhaps because it is measuring the 20% of the energy flow rate from the surface due to emitted infra-red and adding a large amount of infra-red radiation emitted by the atmosphere in the lower troposphere.  In that case, it is to be expected that the measured energy flux will appear to be much amplified.  More likely it is just because the creators of this energy budget believe that every gray body emits radiations just as it would in vacuum with only 0 K bodies surrounding it.  One can make that error and perform many types of calculations to get the right answer.  In no case is it plausible that the long-wave infra-red energy flow rate can exceed 71% in this Fig. 1. Energy Budget unless there is another, non-solar radiation, source of energy.  If the flow of energy through the atmosphere is taken into account, one can show that the flow of energy out of the surface is 48%.

It is not enough to conserve the power density fluxes.  Power density is an energy flow per unit time and per unit area.  That energy must still be conserved as a critical constraint.  One can make up many a mythical power density flow pattern with flux equilibrium across boundaries which violates Conservation of Energy.  This is a frequent problem that brings headaches to many electromagnetic field calculations in physics.  James Clerk Maxwell was adamant that one had to constantly check for Energy Conservation when doing electromagnetic field calculations.  The Stefan - Boltzmann Equation applied to one body in space or to many-bodies is just such an electromagnetic field calculation.

The so-called Energy Budgets are usually a picture of the various energy absorption and emission events that occur, excepting back-radiation, but they are not usually consistent with energy conservation or the conservation of energy flows.  Consequently, the so-called Energy Budgets commonly make it appear that the energy of the original energy packet has somehow been multiplied.  In reality, no such thing has happened.  Energy is conserved.

As a consequence, the Fig. 1 Earth Energy Budget with Surface Radiation of 117% and Back-Radiation of 100% absorbed by the surface is terribly wrong.  The Fig. 3 Kiehl-Trenberth Earth Energy Budget featured in the UN IPCC 4th Assessment Report of 2007 with Surface Radiation of 114% and a down-welling absorption of 94.7% is also terribly wrong.  Both of these and many other Earth Energy Budgets with Surface Radiation Absorbed by the atmosphere (SRA) comparable to or greater than the total absorbed solar insolation in the Earth system are clearly wrong.  So are all those with large or indeed any surface-absorbed down-welling atmospheric radiations (ARAS).  These errors indicate a vast misunderstanding of the role of infra-red absorbing gases in the Earth's climate.  The great exaggeration of the role of so-called greenhouse gases as energy multipliers greatly exaggerates the ability of carbon dioxide to raise the surface temperatures of the Earth as its atmospheric concentration increases.  This is a fatal error in the settled science of catastrophic man-made global warming theory.

For instance, it is commonly stated that a doubling of the carbon dioxide concentration in the atmosphere would cause an additional heating of the surface equivalent to 3.71 W/m², or about 1.1% of solar insolation at the top of the atmosphere.  This claim is actually based on an assumption that all of the warming since the pre-industrial period just after the Little Ice Age was due to the increase in carbon dioxide in the atmosphere over that time.  It is commonly said that this happened because the narrowing of the atmospheric window to long-wave infra-red radiation from the surface caused by higher concentrations of carbon dioxide results in less power density escaping the Earth system into space.  The lowered escape of power density then is said to warm the Earth.  The actual heating due to the narrowing of the atmospheric window is less than is estimated since much of the warming since the Little Ice Age has nothing to do with the increased concentration of carbon dioxide.  When oceans warm, they emit dissolved carbon dioxide in large quantities.  But as is well-known, the oceans warm first and then carbon dioxide concentrations increase over long periods of time, such as 800 years.

At present, the combined effects of water vapor and carbon dioxide do not significantly increase the surface temperature due to adding to the absorption by the surface of infra-red long-wave radiation from the atmosphere.  They add more significantly to the emissivity of the lower troposphere.  The emitted photons then follow the weakening electric field and lower density of infra-red active molecules space-ward, which carries energy into the cooler upper atmosphere and finally into space.  Therefore, the so-called greenhouse gases actually act as coolants in the lower atmosphere, which is where we are most affected by temperature changes.  In addition, increased concentrations of infra-red active gases absorb more solar insolation so that it never reaches the surface to warm it in the first place.  The net effect of increasing their concentrations is a cooling effect of the surface and the lower troposphere, which has long been known through the lowered temperature gradient in the lower troposphere caused by greater humidity.  That effect is dependent upon the fact that the Earth's surface does not absorb down-welling long-wave radiation from the atmosphere in the equilibrium condition in which the atmosphere is cooler than the surface.  The quick energy transport due to radiation emission from the infra-red active gases with comparison to slower convection and water vapor condensation processes, removes heat more rapidly toward higher altitudes, thereby cooling the surface.

However, it should always be remembered that the primary transport mechanisms for heat in the lower troposphere are convection and the water evaporation and condensation cycle.  This is the primary reason the Earth's surface is not in radiative equilibrium with space.  From the standpoint of space, the surface emissivity, as was calculated above, is only about 0.105, while the atmospheric emissivity is about 0.923.  The high collision frequency relative to the time it takes for an infra-red active molecule to re-emit long-wave infra-red radiation absorbed ensures that the active molecule almost always emits infra-red radiation only after it has collided many times with other non-radiating molecules.  Because the mean free paths of such long-wave infra-red radiation as is emitted and absorbed by infra-red active molecules are short, the corresponding temperature differences between the emitting and absorbing molecules are very small.  Therefore, the amount of energy transported by each such process is small.  In addition, these infra-red molecules are relatively rare in the atmosphere, so the total long-wave energy transported by radiation relative to that transported by convection and the water evaporation and condensation processes in the lower troposphere is relatively small. This helps to ensure that radiative effects on the surface temperature, even those of the very dominant infra-red active gas water vapor, are secondary effects, not the primary effects pictured by a claim that greenhouse gases warm the Earth's surface by 33 K.

This post has been updated between 14 March 2014 and 27 June 2014.

[The fact that back-radiation itself is insignificant in explaining the 33C warming of the surface claimed by the proponents of catastrophic AGW is not at all to say that the effects of water vapor on the surface temperature are without significance.  During the day the evaporation of water is an important surface cooling effect and during the night its condensation can be an important warming effect.  During the day, the infra-red active gases absorb a portion of the incoming solar radiation and prevent it from being absorbed by the surface.  At night, fog and clouds scatter, reflect, and absorb the infra-red radiation emitted from the surface.  Also at night, the small true back-radiation effect is not off-set by the absorption of incoming solar radiation by the same infra-red active molecules in the atmosphere.  There is a very important moderation of the surface temperature between day and night due to these effects.  The daily average effect however is small compared to the 33C warming that is commonly claimed by greenhouse gas enthusiasts.  This note was added on 13 November 2014 to remind people of the context in which my criticism of the usual greenhouse gas theory is framed.]