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

In my recent post A Summary of Some of the Physics Errors of the NASA Earth Energy Budget, I discussed a number of problems with the energy budget shown below in which heat transport powers are given as a percentage of the solar insolation at the top of the atmosphere. I demonstrated in that post or referred to earlier posts that demonstrated that the following NASA heat transport power values were very wrong:

1) Back radiation of 100%, which is fictitious when the atmosphere is cooler than the surface

2) Surface Infrared Emission of 117%, which is hugely exaggerated

3) The Surface Absorbed Solar Radiation minus Surface Convection Loss minus Water Evaporation Surface Cooling = 48% - 5% - 25% = 18% in this energy budget. If those values are correct then this sum is also equal to the Surface Infrared Emission. However, I went on to show that this 18% value is too high and/or the 12% of the Earth’s surface radiation emitted through the atmospheric window into space is too low, because it implies that even if the atmosphere were a black body absorber, its temperature would have to be lower than any temperature found in the atmosphere to absorb such a large fraction of the surface emitted infrared radiation.

In conclusion there is virtually no heat transport power in the NASA Earth Energy Budget which is correct. However, given the stridency with which the so-called settled science of the catastrophic man-made global warming hypothesis is said to be unquestionable, let us run through an exercise using some of their own heat transport powers to prove that carbon dioxide at its present atmospheric concentration has a negligible effect on surface temperatures and that increases in that concentration will also have negligible effects on the surface temperature. Remember that this calculation that I will be doing is based on the values provided as an integral and essential part of the so-called consensus science which also is often claimed to be the settled science.

I am going to ask the question what would the surface temperature be if there were no infrared-active gases in the atmosphere, these being the gases commonly called greenhouse gases.  When we have removed all such infra-red active gases from the atmosphere, including carbon dioxide and water vapor and the clouds that result from water vapor, we will calculate an equilibrium average surface temperature.  There will be no surprise in that result.  We will then add carbon dioxide back into the atmosphere and recalculate the surface temperature.  The change in temperature will be the temperature effect of the current carbon dioxide atmospheric concentration on the surface temperature.  This is where the surprise relative to the effect prescribed by the advocates of catastrophic man-made global warming will be seen.  In fact, this result will also surprise the lukewarmers as well.

In order to remove the infra-red active or greenhouse gases, we need to look at another somewhat earlier NASA schematic diagram of heat transport in the atmosphere so that we can separate out the solar insolation reflected from the atmosphere and that part reflected from clouds. We must also be able to use NASA values to separate the portions of the solar insolation absorbed by the atmosphere from that part absorbed by clouds. Of course we expect this alternative NASA Energy Budget to agree with the one above because when the energy budget was promulgated it was already being claimed that the science was settled.

           

Note that 51% is absorbed by the surface instead of 48%, that the sum of the solar insolation reflected from clouds and the atmosphere is 26% instead of 23%, that 19% of solar insolation is absorbed by the atmosphere or by clouds rather than 23%, that solar insolation reflected from the surface is 4% instead of 7%, that conduction is 7% instead of 5%, that water evaporation is 23% instead of 25%, that the surface radiation emitted through the atmospheric window into space is 6% rather than 12%, and that the surface infrared radiation absorbed by the atmosphere is 15% instead of 105%. The second schematic was the viewpoint before that of the first schematic. One has to wonder what the claimed consensus does agree upon and how that agreement is deemed sufficient to make the scientific issues so settled.

If one removes water vapor from the atmosphere, there are no clouds and the mechanism of putting water vapor into the atmosphere has to be removed, namely the evaporation of water. Some solar insolation will still be reflected from the atmosphere or scattered from it with little loss of energy, but most of the reflection is from clouds. The second energy budget tells us that 6%/26% = 0.23 is the fraction of atmospheric reflection not by clouds. The total reflection of both clouds and the atmosphere in the first energy budget is 23%, so the percentage still reflected by the atmosphere with no clouds is (0.23)(23%) = 5.3%. Thus the portion of the solar insolation no longer reflected by clouds is 23% - 5.3% = 17.7%. The fraction of the atmosphere plus clouds absorption of solar insolation which is absorbed by the clouds is 3%/19% = 0.158. The portion of the solar insolation absorbed by clouds is then (0.158)(23%) = 3.6%. Thus the portion of the solar insolation still absorbed with no clouds is 23% - 3.6% = 19.4%.

The removal of the infrared-active gases from the atmosphere also requires us to remove the absorption of incoming solar radiation in the atmosphere by the infrared-active gases. We need additional information to estimate the fraction of the solar insolation absorbed by the infrared absorbing gases. Let us consider these two graphics:

It appears that of the total absorption of solar insolation by the atmosphere, about one-third is absorbed by the infrared-absorbing gases. We just determined that the atmosphere absorbs about 19.4% of solar insolation when we remove clouds, but still have water vapor and other infrared-active gases in the atmosphere. One-third of this is about 6.5%.

Now let us add up the solar insolation power incident upon the surface, P_SI, based on the power values of the first NASA Earth Energy Budget:

P_SI = 55% + 17.7% + 3.6% + 6.5% = 82.8%,

so the solar insolation incident upon the surface when water vapor, clouds, and all infrared-active gases are removed is much increased relative to the current 55% with them present.

At present, 7%/55% is reflected according to the first NASA energy budget above. Let us assume this fraction of reflected solar incident radiation is unchanged, so that the power of solar insolation now absorbed by the surface, P_ABS, is:

P_ABS = (1 - {7%/55%})(82.8%) = 72.3%

This power absorbed by the surface is now going to be dissipated by convection and by radiation. In the NASA Earth Energy Budget, 5% is dissipated by convection. Given that water evaporation is not now occurring, it would be reasonable to think the energy dissipation rate by means of convection might go up, but let us do the calculation with a 5% value. The power radiated by the surface to space, P_SRS, is then

P_SRS = 72.3% - 5% = 67.3%

Now apply the Stefan-Boltzmann Law with an emissivity of 0.95 for the surface and we have

P_SRS = (0.673)(340 W/m²) = (0.95) σ T_S⁴,

where T_S is the surface temperature with no clouds and no greenhouse gases. Solving for T_S,

T_S = 255.30 K

Which is the effective radiation temperature of the Earth today with clouds and infrared-active gases in the atmosphere. So, if the NASA power numbers we used from the NASA Earth Energy Budget are right, then water vapor and clouds and other infrared active gases cause the Earth to be 33K warmer than it would be without them. This is consistent with a common claim of the so-called settled science.

Now I will put carbon dioxide back into our atmosphere and observe the effect of doing so.

Nearly all of the absorption of solar insolation by the atmosphere in the infrared spectrum is due to water vapor. How much is absorbed by carbon dioxide is little addressed by the settled science beyond a determination to ignore this cooling effect. To eye-ball the last figure above, it appears that the absorption by CO₂, is about one-fifteenth that by water vapor. Since the total infrared absorption of solar insolation by the atmosphere was estimated above to be 6.5%, the part due to carbon dioxide is then about one-sixteenth of that or 0.41%.  However, about 7/55 of this is reflected and not absorbed by the surface, leaving about 0.36% less surface absorption.

In the first NASA Earth Energy Budget above, the infrared radiation from the surface absorbed by the atmosphere is given by (117% - 100%) - 12% = 5%, where the difference of the first two powers in parentheses is the real surface infrared emission and 12% is lost through the atmospheric window into space. Most of the 5% of the atmospheric absorption of the surface infrared emission is due to water vapor. It is also important here to only add in that part of the surface emission absorption that carbon dioxide adds to that absorption normally done by water vapor if we would find the relevant net effect of carbon dioxide. From the figure immediately above, this additional CO₂ absorption effect relative to that of water vapor is about one-eighth.  The fraction of the 5% of surface emission absorbed by the atmosphere is then one-ninth of 5% or 0.56%.

I am looking to modify the calculation for the surface temperature that we did above after removing water vapor, clouds, and carbon dioxide from the atmosphere to accommodate the return of carbon dioxide to the atmosphere to see how the surface temperature changes. It is clear that one needs to subtract the insolation power lost to surface absorption due to CO₂ absorbing it in the atmosphere and one wants to add some power due to some additional absorption in the atmosphere of the longwave radiation emitted from the Earth’s surface. Using the full value of the atmospheric absorption of the Earth's infrared emission is actually going to make the warming effect on the surface temperature larger than it is. Most of the heating in the atmosphere stays in the atmosphere and does not cause the surface heat supply to change. It is also likely that the cooling effect on the surface operates at a greater efficiency than does the warming effect.  Consequently, what we will be calculating here is an exaggerated, upper limit on the magnitude of the effect of having CO₂ in the atmosphere and may even have the wrong sign.

Upon subtracting the insolation power lost by CO₂ absorption and adding the power of surface infrared emission absorbed by the atmosphere, we get

P_SRS = (0.673 - 0.0036 + 0.0056)(340 W/m²) = (0.95) σ T_S⁴,

and we must observe that each of the cooling and warming effects of carbon dioxide are already the equivalent of mere rounding errors and the difference between them is still more piddling. Nonetheless, let us carry out the calculation to obtain what is surely an upper limit on the surface temperature:

P_SRS = (0.675)(340 W/m²) = (0.95) σ T_S⁴,

T_S = 255.49 K,

making the total warming effect of carbon dioxide all of about 0.19 K at most at its present concentration. This implies that a doubling of the present CO₂ concentration in the atmosphere will produce a temperature increase much smaller than 0.19 K due to the logarithmic decrease of absorption with increased concentration.  Even at this upper limit of a 0.19 K temperature increase due to the present levels of CO2 in the atmosphere, the fraction of the greenhouse gas warming due to carbon dioxide is only about

0.19 K / (288.15 K - 255.30 K) = 0.0058 or 0.58%

Meaning that the so-called greenhouse effect is about 99.4% attributable to water vapor and the clouds that result from water vapor.

What is more, in the real world in which there is water vapor, there is a negative, not a positive, feedback response by water vapor which would erase a portion of the fraction of 0.19 K that a doubling of the carbon dioxide concentration in the atmosphere would produce according to these approximations using NASA Earth Energy Budget values.

 As I have maintained since 2010, the net effect of carbon dioxide on the surface temperature is entirely negligible. Great increases in the carbon dioxide concentration in the atmosphere will have no significant effect on the surface temperature of the Earth. For all intents and purposes, only water vapor has significant effects on the surface temperature of the Earth and most of its effect is due to clouds and the water evaporation-condensation cycle.  There is no real reason for alarm about the effects of using fossil fuels based on added emissions of CO₂.

It is unfathomable that after governments around the world have spent well over $100 billion on the catastrophic man-made global warming hypothesis that relatively few scientists are pointing out the errors and contradictions that riddle the so-called settled science. It would appear that government funding of science corrupts science absolutely and/or makes scientists incompetent. It is an interesting parallel to what government power does to government employees.

Updated 2 July 2018.

A Summary of Some of the Physics Errors of the NASA Earth Energy Budget

         I have previously discussed many errors in the physics of the NASA Earth Energy Budget which are critical to the argument backing the catastrophic man-made global warming hypothesis. These errors are essentially the same in the Earth Energy Budgets of the UN IPCC reports, though there are minor variations in the values of the heat transport powers in the Earth system consisting of its surface and its atmosphere. The NASA Earth Energy Budget is shown below, where the heat transport is denoted as a percentage of the average solar insolation at the top of the atmosphere:

Among these errors are:

•         The transport of heat in the atmosphere does not address the critical role in the temperature profile played by the action of gravity on air molecules. This is not an actual error in the Earth Energy Budget, but that budget does serve to misdirect attention toward a completely radiation and heat transport dominated view of the problem.

•         The 117% surface radiation from the Earth’s surface requires the Earth’s surface to directly interface to vacuum, with no atmosphere present. The Earth’s surface must be at 289.4 K, be a black body radiator with an emissivity of 1.00, and be surrounded only by space at very nearly 0 Kelvin (K). Note that 289.4 K is a higher temperature than that usually taken to be the Earth’s average surface temperature and that the Earth’s surface emissivity is usually said to be about 0.95. The lack of vacuum at the interface with the Earth’s surface is a serious problem because the surface oscillating dipoles that radiate infrared energy cannot provide that same kinetic energy that creates radiated energy to evaporating water or transfer it to air molecules colliding with the surface. Energy must be conserved. The higher temperature and emissivity used for the surface is a smaller error, but indicative of a cavalier attitude to the science.

•         The Conservation of Energy in a system in equilibrium does not allow the flow of energy into the Earth’s surface to exceed the rate at which energy enters the system. Energy enters this system at 100%, yet this NASA Earth Energy Budget claims it is incident upon the Earth’s surface at a rate of 7% reflected solar insolation plus 48% absorbed solar insolation plus 100% back radiation from the atmosphere for a total of 155%.

•         The atmosphere cannot possibly absorb as much radiation from the surface of the Earth as is claimed to be absorbed, because the atmosphere is not as absorbing as would be a black body absorber and a black body absorber would have to be at a lower temperature than any temperature in the Earth’s atmosphere to absorb as much radiation as the so-called settled science Earth Energy Budget claims is absorbed. This is because the power absorbed by a black body absorber at temperature T_A from a black body emitter at a temperature of T_E at equilibrium is P = σT_E⁴ - σT_A⁴. In the above schematic diagram, it is not possible for the surface to emit 1.17 P_SI, where P_SI is the solar radiation at the top of the atmosphere, and have (1.17 - 0.12) P_SI = 1.05 P_SI be absorbed by the atmosphere. See my discussion of this issue in A Critical Lesson from the NASA Earth Energy Budget.

•         In Solving the Parallel Plane Black Body Radiator Problem and Why the Consensus Science is Wrong, I proved that the consensus science method of applying the Stefan-Boltzmann Law of Thermal Radiation causes the essential characteristic energy density of a black body cavity in equilibrium to double relative the energy density given by Stefan’s Law. Stefan’s Law states that the electric field energy density in a black body cavity is e = aT⁴, where T is the temperature in Kelvin and a is Stefan’s constant. The correct energy density is maintained in the case of two parallel planes at temperatures T_W and T_C with T_W > T_C in the limit that T_C approaches T_W, if the radiation from the warmer plane toward the cooler plane is given by P_W = σT_W⁴ - σT_C⁴ and the radiation from the cooler plane toward the warmer plane is given by P_C = 0. The settled science thinks P_W = σT_W⁴ and P_C = σT_C⁴, which causes there to be many more photons with real energy between the planes than there really are and causes the doubling of the energy density known in Stefan’s Law. Applying this result to the NASA Earth Energy Budget one realizes that there is no equilibrium back radiation from the cooler atmosphere to the warmer surface, so the 100% back radiation is fictitious. Equally important, if the atmosphere were a black body, the radiation from the surface would also be much reduced to the extent that the atmosphere were absorbing some of it. Other critics have made the claim that cooler bodies do not radiate toward warmer bodies using a simple argument based on the Second Law of Thermodynamics, which by itself is not sufficient. However, coupling that law with a minimization of the total energy in the system, which provides the correct result to many a physics problem, does provide a pretty good argument for the same result that I worked out from electromagnetic field thermodynamics. Note that the elimination of back radiation eliminates a power incident upon the surface of 100% and therefore eliminates the violation of the Conservation of Energy at the Earth’s surface discussed in the third bullet above. There are serious consequences of using black body radiation theory in a manner that doubles the energy density of a black body cavity.

Further Discussing the Diminished Role of Radiation in the Lower Atmosphere

          Let us consider the equilibrium condition now at the Earth’s surface that the flow of energy into the surface per unit area must equal the flow of energy out of the surface per unit area. The power absorbed by the surface from solar insolation, P_ABS, according to the NASA Earth Energy Budget is 48%. We now know that the other input to the surface they claim from back radiation is zero in the equilibrium case in which the air cools with increasing altitude from the surface. This is not quite true on average for the real Earth system since there are occasions, commonly in the dawn hours and shortly afterwards, when the air temperature just above the surface is warmer than the surface. This is easily recognized as the cause of dew and ground fog. Consequently, I will allow that back radiation might be 1 or 2%, but the upcoming discussion will ignore this small effect.

          The flow of energy out of the Earth’s surface according to NASA is given by the sums of 5% power lost in convection, 25% power loss through evaporation, and the radiated power P_R. Consequently, we have

P_ABS = (0.48)(340 W/m²) = (0.05 + 0.25)(340 W/m²) + P_R

Solving for P_R, we get

 P_R = (0.18)(340 W/m²)

From the NASA Earth Energy Budget we know that radiation passing through the atmospheric window into space from the surface without any atmospheric absorption is a power, P_AW, of 12% of the top of the atmosphere solar insolation. The remaining power radiated from the Earth’s surface is absorbed by the atmosphere and converted into an upward power loss as convection, R_CC. Thus we have

P_R = P_AW + P_CC

(0.18)(340 W/m²) = (0.12)(340 W/m²) + P_CC

P_CC = (0.06)(340 W/m²)

          Consequently, if NASA has correctly measured the radiation emitted from the surface through the atmospheric window into space, the absorption of solar insolation by the surface, and the sum of the heat loss from the surface due to convection and water evaporation, then the fraction of the radiation from the surface which is absorbed in the atmosphere is only half that of the radiation from the surface that escapes into space without absorption in the atmosphere and it is one-third of the total radiation emitted by the surface. According to the NASA Earth Energy Budget the radiation emitted by the surface of 117% has all but 12% absorbed by the atmosphere, which means that water vapor and carbon dioxide and the various minor infrared-active gases, the greenhouse gases, are playing a huge role in absorbing a power of 105%. In the next to last bulleted item above, I showed that the atmosphere cannot possibly absorb so much infrared radiation from the surface. In reality, we see above that these gases only absorb 6% according to the NASA numbers after we eliminate those that are clearly wrong. The role of infrared-absorbing gases has thus been falsely magnified by a factor of

(105%) / (6%) = 17.5

In light of these observations, is it not interesting that so many are claiming that the science is settled and that there is a scientific consensus that mankind is faced with catastrophic global warming resulting from his generation of carbon dioxide and the use of fossil fuels?

          Given the errors in the science of climate change that I have pointed out here, one should wonder how accurate any of the NASA and the similar values used in the UN IPCC reports might be.

          There is another way in which the NASA Earth Energy Budget is quite misleading with respect to the atmospheric absorption of infrared radiation from the surface. In reality, in most of the world the main part of the surface radiation that is absorbed is absorbed within a very few meters of the surface and not far up into the atmosphere as the diagrams for energy budgets picture the absorption. There are some areas such as the polar regions and a few deserts where the distance for absorption is significant, but in most of the world the humidity is high enough that the absorption length is very short based on laboratory measurements of absorption cross sections or mean free path lengths. Surface radiation in the colder polar regions is substantially less than that from the warmer regions of the Earth, so the longer absorption lengths in those polar regions are also of less importance to the energy budget. That much smaller part of the absorption of surface radiation performed by carbon dioxide is also occurring very close to the surface, though it is a few times greater than the average distance for water vapor, but is also more uniform over the Earth since the concentrations of carbon dioxide in the atmosphere are more uniform.

          If the surface infrared emission is 18% and the atmosphere absorbs 6%, then the temperature a black body absorber in the atmosphere, T_A, would have to be at to absorb so much infrared radiation can be calculated from:

(0.06)(340 W/m²) = (0.18)(340 W/m²) - σT_A⁴

T_A = 163.8 K

This is a temperature lower than that found in the Earth’s atmosphere, so even a black body absorber cannot absorb such a large fraction of the infrared radiation emitted from the Earth’s surface as is implied by the NASA values in the Earth Energy Budget after we have eliminated the errors I pointed out in the bullets at the start of this post. The infrared-active gases can only absorb a fraction of what a black body absorber can, so they certainly cannot remove as large a fraction of the surface-emitted infrared as could a black body absorber.

          I expect the easiest power value for NASA to measure accurately is the 12% surface-emitted radiation through the atmospheric window into space. But, I expect that their measurements of the surface absorption of solar insolation, the loss of surface energy due to convection, and the loss of surface energy due to the evaporation of water are not very well-established numbers. Clearly, the fraction of the surface-emitted infrared energy absorbed by the atmosphere cannot be as high as one-third. NASA has probably substantially underestimated the sum of the heat loss of the surface by means of water evaporation and convection.

          Such is the sad state of the so-called settled science of man-made global warming and such is the foolishness of the scientific consensus on climate change, insofar as that exists.