Water Vapor and Gravity Act Together to Warm the Earth

It is commonly said that greenhouse gases cause the Earth's surface temperature to be 33K warmer than it would be without these infrared-active gases.  This is based on the idea that the total solar insolation presently absorbed by the Earth, both surface and atmosphere, would all be emitted from the surface as thermal radiation and emitted directly to space if there were no greenhouse gases in the atmosphere. Using the Stefan-Boltzmann Law for thermal radiation, the surface temperature is then said to be 255 K.  No correction is made for the fact that without greenhouse gases, the solar radiation lost to space by reflection would be less.  There is no correction either for the fact that most of the solar radiation absorbed by the atmosphere would be absorbed by the surface instead.  Let us make these corrections here to derive a more realistic total greenhouse gas effect as a starting point for understanding how water vapor and gravity together cause the Earth to be as warm as it is.

Reference is made to two NASA Earth Energy Budget diagrams to determine the reflections from the atmosphere, clouds, and the surface:

Despite the long-standing claim that the science is settled, it is interesting to note that these two NASA energy budgets, representing the settled science as they do, do not agree.

The first one says the sum of the solar radiation reflected by the atmosphere and clouds is 23%, while the second one says that sum is 26%.  The first one says the surface reflection is 7%, while the second one says it is 4%.  The first one says that 48% of the incoming solar radiation is absorbed by the surface, while the second one says that 51% is absorbed.  The first NASA energy budget says that 23% of the solar radiation is absorbed by the atmosphere and does not separate out the percentage absorbed by clouds.  The second energy budget says the sum of the incoming solar absorbed by the atmosphere and clouds is 19%.  Let us bear these differences in mind as we look at the following NASA Earth Energy Budget that states that the Earth is warming due to 0.6 W/m² , which is only 0.18% of the incoming solar radiation.  With several percentage differences in the long settled science, it is fascinating that NASA can peer through the uncertainties to see a warming Earth due to only 0.18% of the solar radiation.  Among other things, NASA makes the claim that this 0.18% is not due to changing solar output.

In the third NASA energy budget, the solar radiation reflected by clouds and atmosphere is 22.6% and that reflected from the surface is 6.7%, in close agreement with the first energy budget.  Now the problem is that a portion of the reflection from the atmosphere is due to reflection off nitrogen and oxygen molecules.  Some is also due to reflection off haze due to water vapor.  To estimate the added solar insolation absorbed by the surface when greenhouse gases are eliminated from the atmosphere, it is necessary to know the reflection from haze, which would not exist but for water vapor.  There is no certain value for the added solar radiation incident on the surface and no agreement among the three diagrams on the fraction of that which will be reflected.  Such is the settled science.  I will work with an estimate that 22% of the top of the atmosphere solar radiation is added to that incident on the surface in the case of no greenhouse gases.  I will then assume that of the solar radiation incident on the surface, 12.3% is reflected.  This value is from the third diagram where 22.9/(22.9 + 163.3) = 0.123.

22% of 340.4 W/m² is 74.9 W/m² , which is all but 2.1 W/m² of the 77.0 W/m² reflected into space from the atmosphere in the third diagram.  The 77.1 W/m² adsorbed by the atmosphere in that diagram does not happen with no greenhouse gases either. Consequently, the solar radiation incident on the surface is increased to 338.3 W/m² .  Of this, 12.3% is reflected, so 296.7 W/m² is adsorbed by the surface.  This is 133.4 W/m² more than the 163.3 W/m² adsorbed by the surface in the third NASA Earth energy budget and 56.8 W/m²  more than is now absorbed by both the atmosphere and the surface.  This is no small addition to the absorbed energy, though proponents of the catastrophic man-made global warming hypothesis frequently claim that greenhouse gases have little effect on adsorbed solar radiation.  Some will say that I have no business removing the reflection from clouds, but I cannot understand how one can have clouds without the most important greenhouse gas of all, namely water vapor.

Applying the Stefan-Boltzmann equation we have for the surface temperature with no greenhouse gases:

P (W/m²) = 296.7 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, so T= 269.0 K,

which is only 19.0K less than 288.0 K, not the usually quoted 33K less.  The original greenhouse effect is only 19.0/33.0 = 0.576 times the scale we are usually told it is.  This is one of the means by which it is exaggerated in the settled science.

The role of water vapor is huge.  Gravity creates a gradient in atmospheric gas molecule kinetic energies and therefore in their temperature.  This gravity-induced temperature gradient will not have a large effect on the surface temperature unless there is a large atmospheric emission of energy to space from a substantial altitude.  Water vapor is the primary provider of both the required energy at substantial altitudes in the atmosphere.  It helps that the concentration of water vapor in the atmosphere falls off rapidly in the upper troposphere, so that water vapor with its relatively wide range of radiating frequencies can readily radiate most of the Earth's adsorbed solar radiation from the upper troposphere into space.

Water vapor critically reduces the direct emission of surface emitted thermal radiation to space.  In summary, it does this in three key ways:

• Water vapor absorbs most of the two-thirds of the surface thermal emission of infrared radiation that is absorbed in the atmosphere. Carbon dioxide and the other infrared-active gases also participate in this function as minor partners.

• The creation of water vapor reduces the surface emission of infrared radiation at the surface as a competing surface cooling mechanism by soaking up heat in the process of water evaporation.  This is a result of energy conservation.  Energy that goes into evaporating water cannot also go into photon emission. Carbon dioxide plays no role in this mechanism at all.

• The condensation of water vapor at substantial altitudes in the troposphere provides a large store of energy to be radiated from the upper troposphere into space.  There is no corresponding role for carbon dioxide.

The fact of the large emission of radiation by infrared-active molecules at the top of the troposphere also allows thermals to come into play as a means of reducing surface radiation emission and as a feed of energy to the upper troposphere from whence it can be radiated into space.  It is then this radiation to space from the upper troposphere that determines the temperature of the upper troposphere.  This in turn allows the gravitational temperature gradient to increase the surface temperature relative to the temperature of the upper troposphere from which most of the Earth's thermal radiation into space is emitted.

Without greenhouse gases, the gravity-induced temperature gradient still exists, but the temperature anchor has to be the Earth's surface since it is the only possible source for thermal radiation to space.  The infrared-active gases do provide substantial warming of the Earth's surface, but contrary to the usual claim that it is due simply to absorbing surface emitted thermal infrared in the atmosphere, it is really due to all three of the bulleted mechanisms above.  Only water vapor plays a role in two of the three critical mechanisms and it is very dominant in the remaining mechanism. When this is understood, it is clear that water vapor is a much more dominant actor in the warming of the Earth than is recognized in the general settled science viewpoint.  When only the first bulleted mechanism is said to be the reason for a warmer Earth, carbon dioxide seems to play a bigger role than it really does.

The partnership of water vapor with the temperature gradient in the atmosphere due to gravity provides us with a much warmer Earth.  This partnership is completely lost in the so-called settled science.  To learn more about the gravity induced temperature gradient in the troposphere, see mgh, Not Just Greenhouse Gases, Provides a Warm Earth.  When the water vapor and gravity partnership is properly understood, the very minor role of carbon dioxide is better understood.  When that is understood, the wrongheadedness of catastrophic man-made global warming is understood.

The Simple Physics Explaining the Earth's Average Surface Temperature

According to NASA the Earth's Energy Budget is given by the following diagram here and here. This energy budget is shown in Figure 1.

Fig. 1.  This NASA Earth Energy Budget is based on data over 10 years of observations as of 2009.  As of the time of this writing, it is still the Earth Energy Budget on NASA websites.

NASA says that the Earth emits 239.9 W/m² of longwave infra-red radiation into space.  This implies an effective Earth system radiative temperature of 

P = 239.9 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, 

so T = 255.0K, by application of the Stefan-Boltzmann equation.

Now we can go a step further, since NASA says that 40.1 W/m² of longwave infra-red radiation emitted from the surface alone passes through the atmospheric window without absorption by the atmosphere directly into space.  This allows us to calculate the effective radiative temperature of the atmosphere alone.  Consequently,

P = (239.9 - 40.1) W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴,

so T = 243.7 K, the effective radiative temperature of the atmosphere alone as seen from space.

According to the U.S. Standard Atmosphere Table of 1976, this is the temperature at mid-latitudes at an altitude of 6846 meters by interpolation of table data. This is very close to 7000 meters, where gas molecule parameters are given in the U.S. Standard Atmosphere Table of 1976.

What the NASA Earth Energy Budget critically fails to note is that the Earth's gravity causes a linear temperature gradient in the troposphere, the Earth's lower atmosphere, which was well-known to the American scientists who devised the U.S. Standard Atmosphere tables from the 1950s to the final table of 1976.  The U.S. Standard Atmosphere Table of 1976 is found in many editions of the CRC Handbook of Chemistry and Physics, such as the 71st and 96th Editions. One can still obtain it from the government for a fee.  This gravitational field induced linear temperature gradient was also understood by Prof. Richard Feynmann who discusses it in his Feynmann Lectures late in Vol. 1.  Let me provide a simple explanation of this linear temperature gradient due to gravity and the heat capacity of air molecules.

There is a linear gradient in kinetic energy, E_K, with altitude, since the total energy E of a molecule is given by E = mgh + E_K , with mgh the potential energy and h the altitude.  The temperature of a perfect gas molecule is proportional to its kinetic energy, so an increased kinetic energy at sea level compared to its kinetic energy at 7000 meters altitude means the gas molecule is warmer at sea level. 

The temperature at 7000 meters is established by the fact that that is the effective radiative equilibrium altitude for the atmosphere with space.  We can use this fact to calculate the approximate temperature of the surface due only to the temperature gradient in the troposphere below that due to the effect of gravity alone.  This is an approximation because the atmospheric radiative equilibrium with space is really a range of altitudes primarily weighted over the range of 5000 to 11000 meters in the troposphere at the mid-latitudes approximated by the U.S. Standard Atmosphere Table of 1976. There are also small contributions due to mostly carbon dioxide in the very cold tropopause and the warmer stratosphere.  I want to keep things simple here so that we can develop a good physical feel for the physics without becoming lost in reams of integrations and subsequent computer code to arrive at similar, though more exact, results. Of course, there are other factors than the gravitational temperature gradient that also affect the surface temperature, but we will isolate this one factor and consider the effect of the other factors later.

E_K = (3/2) kT, where E_K is the kinetic energy for a perfect monatomic gas molecule, where k is the Boltzmann constant.  However, the lower atmosphere is made up almost entirely of diatomic molecules, with N₂ and O₂ more than 99% of the atmosphere.  E_K = (5/2) kT for a diatomic perfect or ideal gas molecule and (6/2) kT for a polyatomic molecule with more than two atoms.  This is because a diatomic molecule has rotational kinetic energy around each axis  perpendicular to the bond between the two atoms in the molecule.  There are equal amounts of energy in each of the 5 degrees of freedom of the diatomic molecule.  Molecules such as CO₂ and CH₄ with more than two atoms have 6 degrees of kinetic energy freedom. This allows us to tie the total kinetic energy at an altitude to the translational velocities of molecules given in the U.S. Standard Atmosphere table of 1976 for dry air.  The total kinetic energy of the diatomic molecules making up more than 99% of the lower atmosphere is then 5/3 times the translational kinetic energy.

Conservation of energy for a diatomic gas molecule requires that, where 7000 meters altitude is chosen as a reference altitude, since it is the altitude in effective radiative equilibrium with space:

E_K0 = (5/3) (½ m v₀² ) = E₇₀₀₀ = (5/3)(½ m v₇₀₀₀² ) + mgh,

Where E_K0 is the kinetic and the total energy of the gas molecule at sea level, v₀ is its translational velocity there, E₇₀₀₀ is the total energy at 7000 meters altitude, v₇₀₀₀ is the translational velocity of the gas molecule at 7000 meters altitude, m is the mass of the molecule, g is the gravitational constant at 7000 meters altitude, and h is the altitude, here 7000 m.  From the U.S. Standard Atmosphere table of 1976, the mean gas molecule in the atmosphere has a mass of 28.964 amu or 4.8080 x 10^-26 kg, which is greater than the mass of the most common N₂ molecules and less than the mass of the second most common O₂ molecules.  The gravitational constant at 7000 meters altitude is slightly less than that at sea level and is found in the table to be 9.7851 m/s².  The temperature is 242.7K. The translational velocity of the mean molecule at 7000 meters altitude from the table is 421.20 m/s.  The value of E_K0 is calculated to be approximately 1.040 x 10^-20 Joules per mean molecular weight air molecule at sea level based on the translational velocity at 7000 meters and assuming that we are only calculating the static gravitational effect. 

We can now set the gravitational effect E_K0 kinetic energy into the E_K = (5/2) kT equation and calculate what T should be if there were no other cooling effects, such as the evaporation of water.  Note that air convection is not a net changer of the energy here, except for the effect of volume expansion cooling as the warm air rises and the pressure drops. This temperature gradient exists in the static air, yet there is no flow of heat.

 
The surface temperature due to the action of gravity alone on the atmosphere is found from:

E_K0 = 1.040 x 10^-20 J  = (5/2) kT = (5/2) ( 1.381 x 10^-23 J/K )T

T = 301.2 K

This temperature is actually warmer than the 289.5 K temperature implied by NASA in their Fig. 1. Energy Budget, since they always assume the surface emits as a black body.  With a 398.2 W/m² emission, the implied black body temperature is 289.5 K.  More commonly, the average surface temperature is taken to be about 288 K.

If the Earth's surface were 301.2 K due to the gravity temperature gradient alone, the equivalent black body power density to allow us to compare its effect to other effects in the NASA energy budget would be:

P = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴)(301.2 K)⁴ = 466.6 W/m² 

Note that this is not an actual flow of power to the surface.  The atmospheric temperature gradient due to gravity exists under conditions of energy conservation, with the energy of gas molecules always constant.  There is no heat flow in this effect.

Note that a few critics of the catastrophic man-made global warming hypothesis have calculated a thermodynamic lapse rate due to a rising mass of air with a given heat capacity at constant pressure. The lapse rate due to this is 9.81 K/km, compared to the temperature gradient of 8.36 K/km that I just calculated as due to the Conservation of Energy in the gravitational field.  Either way, the temperature gradient due to gravity is large and requires strong cooling mechanisms to reduce the surface temperature to a mere 288K.  In other words, the problem is not explaining why the surface temperature is 288K instead of 255K, but is instead one of explaining why it is as low as 288K.  The fact that the surface also absorbs solar radiation makes this explanation of the cooling mechanisms even more important.

Let us now consider other inputs and outputs of energy from the NASA Earth Energy Budget of Figure 1.  We will use this effective gravitational power, the power absorbed by the surface from the sun directly, the power loss due to thermals, the power loss due to water evaporation, and the power loss due to surface longwave infra-red emission through the atmospheric window directly into space.

We take the power values for each of these from the NASA energy budget.  We will lump the effect of all other warming or cooling mechanisms, including the Earth's surface radiative emissions absorbed by the atmosphere and any back-radiation into an unknown power flux R.

P = (466.6 + 163.3 -18.4 - 86.4 - 40.1 + R) W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) (288 K)⁴,

Solving for R, we find that 

R = - 94.9 W/m² 

According to NASA in Figure 1, R is equal to the - (358.2 - 340.3) W/m² = - 17.9 W/m² , the difference between the longwave surface emission absorbed by the atmosphere and the back radiation from the atmosphere. This is a modest cooling mechanism for the surface.  What we see from our calculations though is that there is really much more cooling needed than 17.9 W/m² .  

Now there certainly is surface radiation which is absorbed by the atmosphere.  I have maintained that there is very little back radiation and what there is is due to those situations in which the air temperature near the surface is higher than that of the surface.  This is not the common daily condition, but it does happen at times.  I have also maintained that the electric dipoles that oscillate to emit radiation characteristic of a body at 288K cannot provide kinetic energy to the evaporation of water and to transfer to air molecules in collisions.  Energy must be conserved.  Some of the molecules in the surface do emit longwave radiation as though they were part of a surface at 288K, but some do not, because they gave up their energy to the evaporation of water or in collisions with air molecules.  The sum of the radiation energy emitted from those that emit radiation is less than that of a surface at 288K which was not also being cooled by water evaporation and air gas molecule collisions.  For radiation purposes, only a portion of the surface area emits longwave infra-red and this portion is distributed nanoscopically over the entire surface area.  Other nanoscopic areas are dumping their energy instead into evaporating water.  Still others are exchanging it with cooler air molecules which are impinging upon a water surface.  At sea level, the average air molecule has 6.92 x 10⁹/s collisions with other air molecules.  Due to the more than 100 times as many water molecules at the surface of a square meter of water compared to the two-thirds power of the number density of air molecules in a cubic meter, the collision rate for air molecules with water molecules at the surface is more than 100 times greater than that for air molecules with other air molecules.  This gives surface water molecules many opportunities to transfer energy to cooler air molecules.  The story is similar for the other materials of the Earth surface.

The NASA claims that the surface emits a total of 398.2 W/m² and that back radiation amounts to 340.3 W/m² are both absurdly exaggerated.  Surface infra-red emission is not only limited by conservation of energy and the competition with energy loss via water evaporation and gas molecule collisions, but also by the mean free path length of emitted infra-red radiation before absorption of that radiation by water vapor or carbon dioxide molecules.  That limited mean free path length also places a limit on the temperature differential between the surface temperature and the temperature of the air layer where absorption takes place.  The allowed energy transfer is governed by the equation:

P = σ (T_S⁴ – T_A⁴ ), 

where T_S and T_A are the temperatures of the surface and the air layer, respectively.  Since the mean free path length for water vapor absorption of those wavelengths it can absorb is short in the lower troposphere and that for carbon dioxide absorption is not much longer, this is a severe limit on the power transported by radiation from the surface.

The NASA claim that the surface emits 398.2 W/m² includes only 40.1 W/m² emitted directly to space.  The remaining 358.1 W/m² that NASA claims is absorbed by the atmosphere implies that the atmosphere is at a temperature of 0 K!  This could not be more wrong.  On the other hand, a back radiation of 340.3 W/m² implies that the effective temperature of the radiating atmosphere to the surface is 278.3 K.  In the U.S. Standard Atmosphere of 1976, this is an altitude of about 1500 m.  So the atmosphere absorbs radiation as though it were cold space and it emits radiation back to the surface as though it has a temperature of 278.3 K.  The implied mean free paths for the longwave infra-red radiation which can be absorbed and emitted by infra-red active or greenhouse gases are completely incompatible!

What is more, the atmosphere is generally cooler than the surface, so it radiates only to still cooler parts of the atmosphere or to space.  Cooler parts of the atmosphere are usually at higher altitudes, so there is a strong preference for radiated energy from the atmosphere to be transported to higher altitudes.  Such transport to lower altitudes as does occur requires a temperature inversion in air layers.

In any case, it is very clear that the net effect of surface radiation absorbed by the atmosphere, back-radiation, any errors in the solar radiation absorbed by the surface, or power lost as thermals, or power used to evaporate water, or the neglect of other heating or cooling mechanisms, amounts to a net cooling effect of 94.9 W/m² .  This does not leave much room for a warming of the surface attributable to back-radiation from greenhouse gases!

In fact, we need more cooling effects to explain the surface temperature being as low as it is.  A big part of that is surface longwave radiation absorbed by the atmosphere, but I believe another part of that is other cooling mechanisms due to infra-red active gases (greenhouse gases) and underestimation of the cooling by water evaporation and thermals.

Infra-red active gases (water vapor, carbon dioxide) have a higher heat capacity than do nitrogen and oxygen molecules. Consequently, they can remove more energy from the surface upon collisions with it and add to the heat loss of the surface due to rising thermals.  In addition, they make minor contributions to the transfer of heat upward through the atmosphere by radiating infra-red from warmer air layers to the usually cooler air layers just above them.  Since that transport of energy is at the speed of light, this effect transfers energy faster than a rising thermal and acts as a cooling mechanism.

The advocates of catastrophic man-made global warming pose the problem of our average surface temperature as one of explaining why it is as high as it is.  The real problem is explaining why it is as low as it is.  This is the sad state of NASA and generally of the entire set of alarmist man-made global warming advocates understanding of the climate.  For this poor service, NASA now spends more than half of its budget on climate change, with the emphasis heavily on catastrophic man-made global warming.

Update: 5 March 2017