Guest post by Philip Mulholland
The following two figures, showing the principal features of the Earth’s Energy Budget, were published in 1997 by the Oklahoma Climatological Survey (OK-First) and are reproduced here with kind permission.
Both of these diagrams when combined provide detailed energy budget information for the Earth’s climate; however, their parameters are recorded as percentages of solar insolation at the top of the atmosphere (TOA). Neither diagram published by OK-First records the actual values of solar power intensity, nor is it demonstrated how they can be used to estimate the global average temperature for the surface of the Earth.
A number of assumptions must be made in order to understand how the OK-First diagrams can be used to estimate the average global temperature under an expected solar insolation radiant power intensity of *1368 W/m2, and the albedo of 0.30 used in Figure 1. *N.B. The standard NASA Earth irradiance is 1361 W/m2 and the Bond albedo is 0.306 (Williams, 2019). However, in 1997 the solar irradiance used by Kiehl and Trenberth (1997) was 1368 W/m2, and so this value is used here to give the most appropriate match to this historic paper (Fig. 3) (reproduced below with kind permission).
2. Filling in the Gaps.
At first sight it is clear that Figure 1 shows that 30% of the solar insolation is bypassed via albedo loss, and so only 70% of the power intensity is available to heat the planet. If we now apply the standard divide by 4 spherical geometry rule to the expected (but not yet confirmed) solar irradiance of 1368 W/m2, then the TOA power intensity will be reduced to 235 W/m2 post-albedo (as per Kiehl and Trenberth, 1997). However, and confusingly, because the percentages relate to the unfiltered TOA power intensity, it follows that the power intensity values in the OK-First diagrams are percentages of the assumed (but not yet confirmed) pre-albedo value of 342 W/m2, and so this power intensity number must be used. By this means consistency in both percentages and also power intensity values will be maintained throughout the OK-First diagrams, the elements of which are presented below in Table 1.
The next assumption we must make is that the standard partition of energy by the atmosphere is being applied. The standard assumption is that for all energy fluxes intercepted by the atmosphere, half of the flux is directed upwards, and lost to space, and half of all captured flux is returned to the surface as back radiation and recycled. This concept is shown in figure 4 (reproduced here with kind permission).
Fig. 4: Equipartition of energy flux by the Atmospheric layer (Jacob, 1999 Fig.7-12)
Because the intercepted energy flux is being recycled this feed-back loop is an endless sum of halves of halves. It has the mathematical form of a geometric series, and is a sum of the descending fractions in the power sequence 2– n, where minus n is a continuous sequence of natural numbers ranging from zero to infinity.
Equation 1: 1/2 + ¼ + 1/8 + 1/16 + 1/32 + …. + 2-n = 1
Equation 1 describes the cumulative effect of the feed-back loop (after an infinite series of additions), where for each turn of the cycle, half the ascending energy flux is passed out to space and lost, and the other half is returned back to the ground surface and then re-emitted. It is a feature of this form of an infinite series that the sum of the series is not itself an infinite number, but in this case the limit is the finite natural number 1.
As a direct consequence of applying Equation 1 to the OK-First atmospheric model we must double the energy flux within the atmosphere, because the atmosphere retains and stores an energy flux equal to that of the total intercepted flux. When we apply the logic of the 50%:50% atmospheric energy flux partition to the OK-First analysis, then we are able to create the following table of percentage atmospheric energy recycling (Table 2): –
Table 2 demonstrates that the power intensity experienced by the atmosphere is 128% of the incoming solar beam, and in addition the power intensity flux emitted by the surface, and directly attributable to the high frequency solar insolation, adds another 51% to the planetary energy budget. This means that the total power intensity flux that drives the Earth’s climate is 179% of the pre-albedo TOA insolation according to the OK-First diagram.
In order to justify what is clearly a contentious statement I will now apply the identical process of deconstruction to the accepted diagram of Kiehl and Trenberth, with its recorded power intensity values (Fig. 3), and compare this with the atmospheric absorption elements as listed in Figs. 1 & 2 by OK-First.
Table 3 demonstrates that the total power intensity flux absorbed by the atmosphere in the Kiehl and Trenberth diagram is 195 W/m2, and that this power intensity is then doubled to 390 W/m2 by the process of atmospheric recycling, which includes recycling of both the thermals and also evaporation energy fluxes. Using the standard Stefan-Boltzmann equation to convert irradiance power intensity to thermodynamic temperature
Where j* is the black body radiant emittance in Watts per square metre, then the average temperature of the Earth’s atmosphere for a total atmospheric power intensity flux of 390 W/m2 is 288 Kelvin (15o Celsius).
Table 4 below demonstrates that the total energy budget for the Earth is driven by 168 W/m2 of surface intercepted and incoming atmospheric absorbed solar insolation. This flux must be added to the intercepted and recycled atmospheric flux of 390 W/m2 (that contains the direct atmospheric solar interception of 67 W/m2) to give a planetary energy budget of 558 W/m2, which equates to a thermodynamic temperature of 315 Kelvin (42o Celsius). The surface fluxes of 1. Surface Longwave Radiation, 2. Thermals and 3. Evaporation are all losses that create surface cooling and so combine to produce the expected Surface Radiation flux of 390 W/m2, which equates to a thermodynamic temperature of 288 Kelvin (15o Celsius).
If at this point you are beginning to wonder why the much-vaunted back radiation has been adjusted, and why some of the returning radiant flux in the Kiehl and Trenberth diagram can be replaced with recycled energy fluxes from the descending air (returned thermals) then please bear with me.
Let us return to the OK-First diagrams (Figs. 1 & 2) now that the table of flux values has been validated using the Kiehl and Trenberth power intensity metrics and apply the same TOA input flux of 342 W/m2 used in Fig. 3 to the table of percentages created from the OK-First diagrams and displayed in Table 2.
This insolation power intensity flux of 342 W/m2, when combined with the published percentages of OK-First can be used to create a table of power intensity values (Table 6) and associated thermodynamic temperatures (Table 7).
The global average surface temperature of 23oC calculated using the OK-First data is higher than that calculated by Kiehl and Trenberth. This temperature difference arises from a number of possible causes.
1. The OK-First model is using a lower Bond albedo.
2. The solar irradiance used by OK-First for the calculation of percentages is unknown but assumed to be the same number as that used by Kiehl and Trenberth.
3. The balance of energy partition fluxes within the OK-First model is different from the canonical model, and this is the most likely cause of the bias towards the calculated higher global average temperature.
Kiehl and Trenberth and OK-First, use identical concepts in the formation of their global energy budget diagrams, however both originators present their results in ways that do not clearly demonstrate the commonality or the rigor of the concepts used. In particular both sources fail to illustrate the implicit role of atmospheric mass movement in the process of energy recycling that also heats the surface of our planet. In the presence of a gravity field that binds the atmosphere to the surface of a planet, what goes up must come down. The distribution of energy fluxes in Table 3 show that for the total atmospheric energy budget of 558 W/m2 (Table 4), 63.44% (354 W/m2) is transmitted by radiation fluxes and 36.56% (204 W/m2) is carried by mass motion (Table 8).
So clearly mass motion is an important energy carrying process within the Earth’s atmosphere. It is critical to understand at this point that because our energy budget is formulated in terms of power intensity, if the proportion of flux carried by mass motion increases due to an increase in moist convectional overturning, then the proportion of energy transmitted by radiant processes must decrease (and vice versa), a given energy flux cannot do two things at once.
In addition, we find that because the energy budgets of OK-First and also Kiehl and Trenberth are clearly built on the equipartition of energy by the atmosphere (half up and half down), then there are only two ways that the internal energy budget of the Earth’s atmosphere can be increased.
1. The longwave surface to space atmospheric window is closed, which causes more energy to be recycled within the atmosphere.
2. The planetary Bond albedo is decreased which allows more solar energy to enter the climate system.
Issue #1 relates directly to concerns that carbon dioxide emissions increase the opacity of our semi-transparent atmosphere, and will close the atmospheric window (Fig. 5). We can test the effects of closing this window on global average temperature by using Table 3, and diverting the 40 W/m2 direct to space radiant emission into atmospheric capture and heating (Table 9).
The impact of closing the Earth’s long wave emission atmospheric window is to raise the global average temperature from 15oC to 29oC (Table 10). This 14oC increase is the maximum possible temperature increase that the Earth can experience by internal energy recycling for a constant Bond albedo of 0.306.
In order to further raise the Earth’s average global temperature above 29oC to form a Cretaceous hothouse world it is necessary to either increase the atmospheric mass, (thereby raising atmospheric pressure and also the boiling point of water), and/or reduce the planetary brightness by lowering the Earth’s Bond albedo. Assuming total blocking of the atmospheric thermal radiant window and also assuming no increase in atmospheric mass, then it is possible to achieve a Cretaceous global average temperature of 36oC with a planetary Bond albedo of 0.244 (Table 11).
This reduction in planetary brightness can be achieved by having a Cretaceous world with no surface icecaps, and also an increased continental surface inundation associated with a high global sea level to create a putative low albedo hothouse world (Table 12).
Replacing reflective continental solid land surfaces with a liquid surface of shallow solar energy absorbing seas means that the Earth would capture and transmit more solar energy from the tropics to the poles via the oceanographic currents of a flooded world (e.g. the Tethys Ocean). Assuming a Cretaceous meteorological distribution of energy flux, pro-rata to that of the modern world, then the key energy budget metrics for a 36oC world are speculatively recorded in Table 13.
There are some fundamental messages that come from this analysis of these diagrams of the Earth’s energy budget: –
Issue #1. Internal energy recycling limits the maximum possible temperature rise to an increase of plus 14oC, assuming total blocking of the longwave atmospheric window and an unchanged Bond albedo. It is impossible for the Earth to experience a runaway greenhouse effect if the total mass of the atmosphere does not increase.
In order to achieve a putative Cretaceous global average temperature of 36oC, it is necessary to both reduce the Earth’s albedo to 0.244, and also to apply total blocking of surface to space longwave radiation (and/or raise the total mass of the atmosphere).
Total blocking of the atmospheric window by Carbon Dioxide may not be possible. This is an issue that was studied by Ferenc Miskolczi (2010) in his paper “The Stable Stationary Value of the Earth’s Global Average Atmospheric Planck-Weighted Greenhouse-Gas Optical Thickness”.
Miskolczi stated his conclusions as: –
New relationships among the flux components have been found and are used to construct a quasi-all-sky model of the earth’s atmospheric energy transfer process. In the 1948-2008 time period the global average annual mean true greenhouse-gas optical thickness is found to be time-stationary. Simulated radiative no-feedback effects of measured actual CO2 change over the 61 years were calculated and found to be of magnitude easily detectable by the empirical data and analytical methods used.
The data negate increase in CO2 in the atmosphere as a hypothetical cause for the apparently observed global warming. A hypothesis of significant positive feedback by water vapor effect on atmospheric infrared absorption is also negated by the observed measurements. Apparently major revision of the physics underlying the greenhouse effect is needed.
Issue #2. Changes in the value of the Earth’s planetary Bond albedo are a valid mechanism by which global warming can occur. Variations in water distribution in the forms of either reflective ice and/or cloud; or absorbing surface water areal variations by either short term sea-ice distribution or long-term geologic ocean distribution (e.g. The Tethys Ocean) is the primary route to change planetary albedo. This dominance of water either in its reflective role of clouds and ice leading to planetary albedo increase, or in its absorptive form as a transparent surface liquid replacing polar sea ice, means that there is no albedo role for atmospheric carbon dioxide to change global average temperatures. Unlike water, carbon dioxide is not a condensing gas in the Earth’s atmosphere, and so it has no impact on insolation energy capture via changes in reflective planetary brightness.
Issue #3. The standard climate model has the following basic features with specific rules applied.
1. The planetary disc intercept rule. – The average solar irradiance is divided by 4 and spread over the surface of the globe.
2. The albedo bypass rule. – A given percentage of the planetary insolation is bypassed by planetary brightness and not used within the climate system.
3. The remaining solar insolation is absorbed by the planet/atmosphere.
4. The planetary atmosphere is leaky. – Low frequency thermal radiation can pass from the surface directly out to space.
5. The atmosphere is an energy reservoir.
6. Energy recycling by the atmosphere doubles the quantity of energy in this reservoir. – The half in / half out rule of back radiation energy flux partition.
7. Rule six limits the maximum possible gain to times 2. –The infinite recycling geometric series limit.
What this all means is that for a planet with a zero albedo surface (that is with 100% insolation high-energy absorption under a totally clear atmosphere) and a totally opaque atmosphere for exiting surface thermal radiation (that is no surface leaks to space and total 100% atmospheric thermal radiant blocking) then the absolute limit of the internal energy budget is 3 times the Solar Irradiance flux divided by 4.
For planet Earth, with a planetary solar irradiance of 1361.0 W/m2 (Williams, 2019), the maximum possible planetary energy budget for a hypothetical Bond albedo of zero and total atmospheric insolation clarity is 1361*0.75 = 1020.75 W/m2. This flux translates into a maximum possible energy budget thermodynamic temperature of 366.3 Kelvin (93.3oC) (Table 14).
For Venus, with a solar irradiance of 2601.3 W/m2 (Williams, 2018), the maximum possible planetary energy budget for a hypothetical Bond albedo of zero and total atmospheric insolation clarity is 2601.3*0.75 = 1951 W/m2. This flux translates into a maximum possible energy budget thermodynamic temperature of 430.7 Kelvin (157.7oC), but the surface temperature of Venus is 737 Kelvin (464oC) (Williams, 2018).
From this analysis we can deduce that the standard climate model is compromised. The back-radiation concept cannot explain why Venus has a surface temperature of 464oC by atmospheric radiant energy flux recycling. The solar flux captured by the Venusian atmosphere is far too low to produce the observed surface temperature, even if that planet had a Bond albedo of zero and total atmospheric insolation clarity (which it clearly does not have).
Jacob, D.J. 1999. Introduction to Atmospheric Chemistry. Princeton University Press.
Kiehl, J.T and K.E. Trenberth, 1997. Earth’s Annual Global Mean Energy Budget. Bulletin of the American Meteorological Society, Vol. 78 (2), 197-208.
Miskolczi, F.M., 2010. The stable stationary value of the earth’s global average atmospheric Planck-weighted greenhouse-gas optical thickness. Energy & Environment, 21(4), pp.243-262.
Oklahoma Climatological Survey 1997 Earth’s Energy Budget.
Williams, D.R. 2018 Venus Fact Sheet.
Williams, D.R. 2019 Earth Fact Sheet.