Foreword – I’ve had this document since January 17th, and it has taken some time to get it properly reproduced here in full due to formatting issues. Some equations have to be converted to images, and I have to double check every superscript, subscript, and symbol for accuracy, then re-insert/re-format many manually since they often don’t reproduce properly in WordPress. WordPress doesn’t manage copy/paste of complex documents well. I hope that I have everything correctly reproduced, if not, please leave a note. A PDF of the original is here: UTC_Blog_Reply_Part1 This is a contentious issue, and while it would be a wonderful revelation if it were proven to be true, I personally cannot see any way it can surmount the law of conservation of energy. That view is shared by others, noted in the opening paragraph below. However, I’m providing this for the educational value it may bring to those who can take it all in and discuss it rationally, with a caution – because this issue is so contentious, I ask readers to self-moderate so that the WUWT moderation team does not have to be heavy handed. I invite you take it all in, and to come to your own conclusion. Thank you for your consideration. – Anthony

**Part 1: Magnitude of the Natural ‘Greenhouse’ Effect **

Ned Nikolov, Ph.D. and Karl Zeller, Ph.D.

**Introduction**

Our recent paper “*Unified Theory of Climate: Expanding the Concept of Atmospheric Greenhouse Effect Using Thermodynamic Principles. Implications for Predicting Future Climate Change*” spurred intense discussions at WUWT and Tallbloke’s Talkshop websites. Many important questions were raised by bloggers and two online articles by Dr. Ira Glickstein (here) and Dr. Roy Spencer (here). After reading through most responses, it became clear to us that that an expanded explanation is needed. We present our reply in two separate articles that address blog debate foci as well as key aspects of the new paradigm.

Please, consider that understanding this new theory requires a shift in perception! As Albert Einstein once noted, a new paradigm cannot be grasped within the context of an existing mindset; hence, we are constrained by the episteme we are living in. In that light, our concept requires new definitions that may or may not have exact counterparts in the current Greenhouse theory. For example, it is crucial for us to introduce and use the term* Atmospheric Thermal Effect* (*ATE*) because: (*a*) The term *Greenhouse Effect* (*GE*) is inherently misleading due to the fact that the free atmosphere, imposing no restriction on convective cooling, does not really work as a closed greenhouse; (*b*) ATE accurately coveys the physical essence of the phenomenon, which is the *temperature boost* at the surface due to the presence of atmosphere; (*c*) Reasoning in terms of ATE vs. GE helps broaden the discussion beyond radiative transfer; and (*d*) Unlike GE, the term *Atmospheric Thermal Effect* implies *no* underlying physical mechanism(s).

We start with the undisputable fact that the atmosphere provides *extra warmth* to the surface of Earth compared to an *airless* environment such as on the Moon. This prompts two basic questions: (1) *What is the magnitude of this extra warmth, i.e. the size of ATE *? and (2) *How does the atmosphere produce it, i.e. what is the physical mechanism of ATE *? In this reply we address the first question, since it appears to be the crux of most people’s difficulty and needs a resolution before proceeding with the rest of the theory (see, for example, Lord Monckton’s WUWT post).

**Magnitude of Earth’s Atmospheric Thermal Effect**

We maintain that in order to properly evaluate ATE one must compare Earth’s average near-surface temperature to the temperature of a spherical celestial body with* no* atmosphere at the same distance from the Sun. Note that, we are not presently concerned with the composition or infrared opacity of the atmosphere. Instead, we are simply trying to quantify the *overall* effect of our atmosphere on the surface thermal environment; hence the comparison with a similarly illuminated *airless* planet. We will hereafter refer to such planet as an equivalent Planetary Gray Body (PGB).

Since temperature is proportional (linearly related) to the internal kinetic energy of a system, it is theoretically perfectly justifiable to use *mean*global surface temperatures to quantify the ATE. There are two possible indices one could employ for this:

- The absolute difference between Earth’s mean temperature (
*T*_{s}) and that of an equivalent PGB (*T*_{gb}), i.e. ATE =*T*_{s}–*T*_{gb}; or

- The ratio of
*T*_{s}to*T*_{gb}. The latter index is particularly attractive, since it normalizes (standardizes) ATE with respect to the top-of-atmosphere (TOA) solar irradiance (*S*_{o}), thus enabling a comparison of ATEs among planets that orbit at various distances from the Sun and receive different amounts of solar radiation. We call this non-dimensional temperature ratio a*Near-surface Thermal Enhancement*(ATEn) and denote it by*N*_{TE}=*T*_{s}/*T*_{gb}. In theory, therefore,*N*_{TE}should be equal or greater than 1.0 (*N*_{TE}≥ 1.0). Please, note that ATEn is a*measure*of ATE.

It is important to point out that the current GE theory measures ATE not by temperature, but by the amount of absorbed infrared (IR) radiation. Although textbooks often mention that Earth’s surface is 18K-33K warmer than the Moon thanks to the ‘greenhouse effect’ of our atmosphere, in the scientific literature, the *actual* effect is measured via the *amount* of outgoing infrared radiation *absorbed* by the atmosphere (e.g. Stephens et al. 1993; Inamdar & Ramanathan 1997; Ramanathan & Inamdar 2006; Houghton 2009). It is usually calculated as a difference (occasionally a ratio) between the total average infrared flux emanating at the surface and that at the top of the atmosphere. Defined in this way, the average atmospheric GE, according to satellite observations, is between 157 and 161 W m^{-2} (Ramanathan & Inamdar 2006; Lin et al. 2008; Trenberth et al. 2009). In other words, the current theory uses *radiative* flux units instead of *temperature* units to quantify ATE. This approach is based on the preconceived notion that GE works by *reducing* the rate of surface infrared cooling to space. However, measuring a phenomenon with its *presumed* *cause* instead by its *manifest effect* can be a source of major confusion and error as demonstrated in our study. Hence, we claim that the proper assessment of ATE depends on an accurate estimate of the mean surface temperature of an equivalent PGB (*T*_{gb}).

**Estimating the Mean Temperature of an Equivalent Planetary Gray Body**

There are two approaches to estimate *T*_{gb} – a theoretical one based on known physical relationships between temperature and radiation, and an empirical one relying on observations of the Moon as the closest natural gray body to Earth.

According to the Stefan-Boltzmann (SB) law, any physical object with a temperature (*T*, ^{o}K) above the absolute zero emits radiation with an intensity (*I*, W m^{-2}) that is proportional to the 4^{th} power of the object’s absolute temperature:

where *ϵ* is the object’s thermal emissivity/absorptivity (0 ≤ *ϵ* ≤ 1 ), and *σ* = 5.6704×10^{-8} W m^{-2} K^{-4} is the SB constant. A theoretical blackbody has *ϵ* = 1.0, while real solid objects such as rocks usually have *ϵ* ≈ 0.95. In principle, Eq. (1) allows for an accurate calculation of an object’s equilibrium temperature given the amount of *absorbed* radiation by the object, i.e.

The *spatially* *averaged* amount of solar radiation absorbed by the Earth-Atmosphere system (*S** _{α}* ̅̅̅, W m

^{-2}) can be accurately computed from TOA solar irradiance (

*S*

*̅̅̅, W m*

_{α}^{-2}) and planetary albedo (

*α*) as

_{p}where the TOA shortwave flux (W m^{-2}) incident on a plane *perpendicular* to the solar rays. The factor ¼ serves to distribute the solar flux incident on a flat surface to a sphere. It arises from the fact that the surface area of a sphere (4*πR*^{2}) is 4 times larger than the surface area of a disk (*πR*^{2}) of the same radius (*R*). Hence, it appears logical that one could estimate Earth’s average temperature in the absence of ATE from using the SB law. i.e.

Here (T_{e}K) is known as the *effective* *emission* temperature of Earth. Employing typical values for *S _{0} *=W m

^{-2}and

*α*= 0.3 and assuming,

_{p}*ϵ*= 1.0 Eq. (3) yields 254.6K. This is the basis for the widely quoted 255K (-18C)

*mean*surface temperature of Earth

*in the*

*absence*of a ‘greenhouse effect’, i.e. if the atmosphere were missing or ‘completely transparent’ to IR radiation. This temperature is also used to define the so-called

*effective emission height*in the troposphere (at about 5 km altitude), where the bulk of Earth’s outgoing long-wave radiation to space is assumed to emanate from. Since Earth’s mean surface temperature is 287.6K (+14.4C), the present theory estimates the size of ATE to be 287.6K – 254.6K = 33K. However, as pointed out by other studies, this approach suffers from a serious logical error. Removing the atmosphere (or even just the water vapor in it) would result in a much lower planetary albedo, since clouds are responsible for most of Earth’s shortwave reflectance. Hence, one must use a different albedo (

*α*) in Eq. (3) that

_{p}*only*quantifies the actual surface reflectance. A recent analysis of Earth’s global energy budget by Trenberth et al. (2009) using satellite observations suggests

*α*≈ 0.12. Serendipitously, this value is quite similar to the Moon bond albedo of 0.11 (see Table 1 in our original paper), thus allowing evaluation of Earth’s ATE using our natural satellite as a suitable PGB proxy. Inserting= 0.12 in Eq. (3) produces

_{p}*T*= 269.6K, which translates into an ATE of only 18K (i.e. 287.6 – 269.6 = 18K).

_{e}In summary, the current GE theory employs a simple form of the SB law to estimate the magnitude of Earth’s ATE between 18K and 33K. The theory further asserts that the Moon average temperature is 250K to 255K despite the fact that using the correct lunar albedo (0.11) in Eq. (3) produces ≈270K, i.e. a15K to 20K higher temperature! Furthermore, the application of Eq. (3) to calculate the *mean* temperature of a sphere runs into a fundamental *mathematical* *problem* caused by Hölder’s inequality between non-linear integrals (e.g. Kuptsov 2001). What does this mean? Hölder’s inequality applies to certain non-linear functions and states that, in such functions, the use of an arithmetic *average* for the independent (input) variable will *not* produce a correct *mean* value of the dependent (output) variable. Hence, due to a non-linear relationship between temperature and radiative flux in the SB law (Eq. 3) and the variation of absorbed radiation with latitude on a spherical surface, one *cannot* correctly calculate the mean temperature of a unidirectionally illuminated planet from the* *amount of *spatially averaged *absorbed radiation defined by Eq. (2). According to Hölder’s inequality, the temperature calculated from Eq. (3) will *always* be significantly *higher* than the *actual* mean temperature of an airless planet. We can illustrate this effect with a simple example.

Let’s consider two points on the surface of a PGB, P_{1} and P_{2}, located at the exact same latitude (say 45^{o}N) but at opposite longitudes so that, when P_{1} is fully illuminated, P_{2} is completely shaded and vice versa (see Fig. 1). If the PGB is orbiting at the same distance from the Sun as Earth and solar rays were the only source of heat to it, then the equilibrium temperature at the illuminated point would be (assuming a solar zenith angle* θ* = 45^{o}), while the temperature at the shaded point would be *T*_{2} = 0 (since it receives no radiation due to cos*θ* < 0). The *mean* temperature between the two points is then *T*_{m} = (*T*_{1} + *T*_{2})/2 = 174.8K. However, if we try using the *average* *radiation* absorbed by the two points W m^{-2} to calculate a mean temperature, we obtain = 234.2K. Clearly, *T*_{e} is *much* greater than *T*_{m} (*T*_{e} ≫ *T*_{m}), which is a result of Hölder’s inequality.

**Figure 1.** Illustration of the effect of Hölder’s inequality on calculating the mean surface temperature of an airless planet. See text for details.

The take-home lesson from the above example is that calculating the *actual mean* temperature of an airless planet requires *explicit integration* of the SB law over the planet surface. This implies *first* taking the 4^{th} root of the absorbed radiative flux at each point on the surface and *then* averaging the resulting temperature field rather than trying to calculate a mean temperature from a spatially averaged flux as done in Eq. (3).

Thus, we need a new model that is capable of predicting* **T*_{gb} more robustly than Eq. (3). To derive it, we adopt the following reasoning. The equilibrium temperature at any point on the surface of an airless planet is determined by the incident solar flux, and can be approximated (assuming uniform albedo and ignoring the small heat contributions from tidal forces and interior radioactive decay) as

where is the solar zenith angle (radian) at point , which is the angle between solar rays and the axis normal to the surface at that point (see Fig. 1). Upon substituting , the planet’s *mean* temperature () is thus given by the spherical integral of , i.e.

Comparing the final form of Eq. (5) with Eq. (3) shows that *T*_{gb} << *T*_{e} in accordance with Hölder’s inequality. To make the above expression physically more realistic, we add a small constant *C*_{s} =0.0001325 W m^{-2} to *S*_{o}, so that when *S*_{o} = 0.0, Eq. (5) yields *T*_{gb} = 2.72K (the irreducible temperature of Deep Space), i.e:

In a recent analytical study, Smith (2008) argued that Eq. (5) only describes the mean temperature of a non-rotating planet and that, if axial rotation and thermal capacity of the surface are explicitly accounted for, the average temperature of an airless planet would approach the effective emission temperature. It is beyond the scope of the current article to mathematically prove the fallacy of this argument. However, we will point out that increasing the *mean *equilibrium temperature of a physical body always requires a *net* input of *extra* energy. Adding axial rotation to a stationary planet residing in a vacuum, where there is no friction with the external environment does *not* provide any *additional* heat energy to the planet surface. Faster rotation and/or higher thermal inertia of the ground would only facilitate a more efficient spatial distribution of the absorbed solar energy, thus increasing the *uniformity* of the resulting temperature field across the planet surface, but could *not* affect the *average* surface temperature. Hence, Eq. (6) correctly describe (within the assumption of albedo uniformity) the global mean temperature of any airless planet, be it rotating or non-rotating.

Inserting typical values for Earth and Moon into Eq. (6), i.e. *S*_{o} = 1,362 W m-2, *α*_{o} = 0.11, and *ϵ* = 0.955, produces *T*_{gb} = 154.7K. This estimate is about 100K *lower* than the conventional black-body temperature derived from Eq. (3) implying that Earth’s ATE (i.e. the GE) is *several times* *larger* than currently believed! Such a result, although mathematically justified, requires independent empirical verification due to its profound implications for the current GE theory. As noted earlier, the Moon constitutes an ideal proxy PGB in terms of its location, albedo, and airless environment, against which the thermal effect of Earth’s atmosphere could be accurately assessed. Hence, we now turn our attention to the latest temperature observations of the Moon.

**NASA’s****Diviner****Lunar Radiometer Experiment**

In June 2009, NASA launched its Lunar Reconnaissance Orbiter (LRO), which carries (among other instruments) a Radiometer called* Diviner*. The purpose of Diviner is to map the temperature of the Moon surface in unprecedented detail employing measurements in 7 IR channels that span wavelengths from 7.6 to 400 μm. Diviner is the first instrument designed to measure the full range of lunar surface temperatures, from the hottest to the coldest. It also includes two solar channels that measure the intensity of reflected solar radiation enabling a mapping of the lunar shortwave albedo as well (for details, see the *Diviner* Official Website at http://www.diviner.ucla.edu/).

Although the Diviner Experiment is still in progress, most thermal mapping of the Moon surface has been completed and data are available online. Due to time constraints of this article, we did not have a chance to analyze Diviner’s temperature data ourselves. Instead, we elected to rely on information reported by the Diviner Science Team in peer-reviewed publications and at the Diviner website.

Data obtained during the LRO commissioning phase reveal that the Moon has one of the most extreme thermal environments in the solar system. Surface temperatures at low latitudes soar to 390K (+117C) around noon while plummeting to 90-95K (-181C), i.e. almost to the boiling point of liquid oxygen, during the long lunar night (Fig. 2). Remotely sensed temperatures in the equatorial region agree very well with direct measurement conducted on the lunar surface at 26.1^{o }N by the Apollo 15 mission in early 1970s (see Huang 2008). In the polar regions, within permanently shadowed areas of large impact craters, Diviner has measured some of the coldest temperatures ever observed on a celestial body, i.e. down to 25K-35K (-238C to -248C). It is important to note that planetary scientists have developed detailed process-based models of the surface temperatures of Moon and Mercury some 13 years ago (e.g. Vasavada et al. 1999). These models are now being successfully validated against Diviner measurements (Paige et al. 2010b; Dr. M. Siegler at UCLA, personal communication).

What is most interesting to our discussion, however, are the *mean* temperatures at various lunar latitudes, for these could be compared to temperatures in similar regions on Earth to evaluate the size of ATE and to verify our calculations. Figure 3 depicts typical diurnal courses of surface temperature on the Moon at four latitudes (adopted from Paige et. al 2010a).

**Figure 2**. Thermal maps of the Moon surface based on NASA’s Diviner infrared measurements showing daytime *maximum* and nighttime *minimum* temperature fields (Source: Diviner Web Site).

**Figure 3**. Typical diurnal variations of the Moon surface temperature at various latitudes. Local time is expressed in lunar hours which correspond to 1/24 of a lunar month. At 89◦ latitude, diurnal temperature variations are shown at summer and winter solstices (adopted from Paige et al. 2010a). Dashed lines indicate annual means at the lunar equator and at the poles.

**Figure 4**. Temperature maps of the South Pole of the Moon and Earth: (**A**) Daytime temperature field at peak illumination on the Moon; (**B**) Nighttime temperature field on the Moon; (**C**) Mean summer temperatures over Antarctica; (**D**) Mean winter temperatures over Antarctica. Numbers shown in bold on panels (**C**) and (**D**) are temperatures in ^{o}K. Panels (**A**) and (**B**) are produced by the *Diviner* Lunar Radiometer Experiment (Paige et al. 2010b). Antarctica maps are from Wikipedia (http://en.wikipedia.org/wiki/Antarctic_climate). Comparison of surface temperatures between Moon’s South Pole and Antarctica suggests a thermal enhancement by the Earth atmosphere (i.e. a ‘Greenhouse Effect’) of about 107K in the summer and 178K in the winter for this part of the Globe.

Figures 4A & 4B display temperature maps of the Moon South Pole during daytime peak illumination and at night (Paige et. al 2010b). Since the Moon has a small obliquity (axial tilt) of only 1.54^{o} and a slow rotation, the average diurnal temperatures are similar to seasonal temperature means. These data along with information posted at the *Diviner* Science webpage indicate that *mean *temperature at the lunar-surface* *ranges from 98K (-175C) at the poles to 206K (-67C) at the equator. This encompasses pretty well our theoretical estimate of 154.7K for the Moon *mean* *global *temperature produced by Eq. (6). In the coming months, we will attempt to calculate more precisely Moon’s actual mean temperature from Diviner measurements. Meanwhile, data published by NASA planetary scientists clearly show that the value 250K-255K adopted by the current GE theory as Moon’s average global temperature is grossly exaggerated, since such high temperature means do *not* occur at *any* lunar latitude! Even the Moon equator is 44K – 49K cooler than that estimate. This value is inaccurate, because it is the result of an improper application of the SB law to a sphere while assuming the wrong albedo (see discussion in Section 2.1 above)!

Similarly, the mean global temperatures of Mercury (440K) and Mars (210K) reported on the NASA Planetary Fact Sheet are also *incorrect*, since they have been calculated from the same Eq. (3) used to produce the 255K temperature for the Moon. We urge the reader to verify this claim by applying Eq. (3) with data for solar irradiance (*S*_{o}) and bond albedo (*α*_{o}) listed on the fact sheet of each planet while setting *ϵ* = 1. This is the reason that, in our original paper, we used 248.2K for Mercury, since that temperature was obtained from the theoretically correct Eq. (6). For Mars, we adopted means calculated from regional data of near-surface temperature and pressure retrieved by the Radio Science Team at Stanford University employing remote observations by the Mars Global Surveyor spacecraft. It is odd to say the least that the author of NASA’s Planetary Fact Sheets, Dr. David R. Williams, has chosen Eq. (3) to calculate Mars’ average surface temperature while ignoring the large body of high-quality direct measurements available for the Red Planet!?

**So, what is the real magnitude of Earth’s Atmospheric Thermal Effect? **

**Table 1**. Estimated Atmospheric Thermal Effect for equator and the poles based on observed surface temperatures on Earth and the Moon and using the lunar surface as a proxy for Earth’s theoretical gray body. Data obtained from Diviner’s Science webpage, Paige at al. (2010b), Figure 4, and Wikipedia:Oymyakon.

**Figure 5**. Earth’s mean annual near-surface temperature according to Wikipedia (Geographic Zones: http://en.wikipedia.org/wiki/Geographical_zone).

Table 1 shows observed mean and record-low surface temperatures at similar latitudes on Earth and on the Moon. The ATE is calculated as a difference between Earth and Moon temperatures assuming that the Moon represents a perfect PGB proxy for Earth. Figure 5 displays a global map of Earth’s mean annual surface temperatures to help the reader visually verify some of the values listed in Table 1. The results of the comparison can be summarized as follows:

**The Atmospheric Thermal Effect, presently known as the natural ****Greenhouse**** Effect, varies from 93K at the equator to about 150K at the poles (the latter number represents an average between North- and South- Pole ATE mean values, i.e. (158+143)/2 =150.5. This range encompasses quite well our theoretical estimate of 133K for the Earth’s overall ATE derived from Eq. (6), i.e. 287.6K – 154.7K = 132.9K. **

Of course, further analysis of the Diviner data is needed to derive a more precise estimate of Moon’s mean surface temperature and verify our model prediction. However, given the published Moon measurements, it is clear that the widely quoted value of 33K for Earth’s mean ATE (GE) is profoundly misleading and wrong!

**Conclusion**

We have shown that the SB Law relating radiation intensity to temperature (Eq. 1 & 3) has been incorrectly applied in the past to predict mean surface temperatures of celestial bodies including Mars, Mercury, and the Moon. Due to Hölder’s inequality between non-linear integrals, the effective emission temperature computed from Eq. (3) is always significantly higher than the actual (arithmetic) mean temperature of an airless planet. This makes the planetary *emission temperature* *T*_{e} produced by Eq. (3) *physically incompatible* with any real measured temperatures on Earth’s surface or in the atmosphere. By using a proper integration of the SB Law over a sphere, we derived a new formula (Eq. 6) for estimating the average temperature of a planetary gray body (subject to some assumptions). We then compared the Moon mean temperature predicted by this formula to recent thermal observations and detailed energy budget calculation of the lunar surface conducted by the NASA Diviner Radiometer Experiment. Results indicate that Moon’s average temperature is *likely* very close to the estimate produced by our Eq. (6). At the same time, Moon measurements also show that the current estimate of 255K for the lunar average surface temperature widely used in climate science is *unrealistically high*; hence, further demonstrating the inadequacy of Eq. (3). The main result from the Earth-Moon comparison (assuming the Moon is a perfect gray-body proxy of Earth) is that the Earth’s ATE, also known as natural Greenhouse Effect, is 3 to 7 times *larger* than currently assumed. In other words, the current GE theory underestimates the extra atmospheric warmth by about 100K! In terms of *relative* thermal enhancement, the ATE translates into *N*_{TE} = 287.6/154.7 = 1.86.

This finding invites the question: How could such a huge (> 80%) thermal enhancement be the result of a handful of IR-absorbing gases that collectively amount to less than 0.5% of total atmospheric mass? We recall from our earlier discussion that, according to observations, the atmosphere only absorbs 157 – 161 W m^{-2} long-wave radiation from the surface. Can this small flux increase the temperature of the lower troposphere by more than 100K compared to an airless environment? The answer obviously is that the observed temperature boost near the surface *cannot* be possibly due to that atmospheric IR absorption! Hence, the evidence suggests that the lower troposphere contains *much more *kinetic energy than radiative transfer alone can account for! The thermodynamics of the atmosphere is governed by the Gas Law, which states that the internal kinetic energy and temperature of a gas mixture is also a function of pressure (among other things, of course). In the case of an isobaric process, where pressure is constant and independent of temperature such as the one operating at the Earth surface, it is the *physical* *force* of atmospheric pressure that can only *fully* explain the observed near-surface thermal enhancement (*N*_{TE}). But that is the topic of our next paper… Stay tuned!

**References**

Inamdar, A.K. and V. Ramanathan (1997) On monitoring the atmospheric greenhouse effect from space. *Tellus 49B*, 216-230.

Houghton, J.T. (2009). *Global Warming: The Complete Briefing (4th Edition)*. Cambridge University Press, 456 pp.

Huang, S. (2008). Surface temperatures at the nearside of the Moon as a record of the radiation budget of Earth’s climate system. *Advances in Space Research* 41:1853–1860 (http://www.geo.lsa.umich.edu/~shaopeng/Huang07ASR.pdf)

Kuptsov, L. P. (2001) Hölder inequality. In: *Encyclopedia of Mathematics*, Hazewinkel and Michiel, Springer, ISBN 978-1556080104.

Lin, B., P. W. Stackhouse Jr., P. Minnis, B. A. Wielicki, Y. Hu, W. Sun, Tai-Fang Fan, and L. M. Hinkelman (2008). Assessment of global annual atmospheric energy balance from satellite observations. *J. Geoph. Res*. Vol. 113, p. D16114.

Paige, D.A., Foote, M.C., Greenhagen, B.T., Schofield, J.T., Calcutt, S., Vasavada, A.R., Preston, D.J., Taylor, F.W., Allen, C.C., Snook, K.J., Jakosky, B.M., Murray, B.C., Soderblom, L.A., Jau, B., Loring, S., Bulharowski J., Bowles, N.E., Thomas, I.R., Sullivan, M.T., Avis, C., De Jong, E.M., Hartford, W., McCleese, D.J. (2010a). The Lunar Reconnaissance Orbiter Diviner Lunar Radiometer Experiment. *Space Science Reviews*, Vol 150, Num 1-4, p125-16 (http://www.diviner.ucla.edu/docs/fulltext.pdf)

Paige, D.A., Siegler, M.A., Zhang, J.A., Hayne, P.O., Foote, E.J., Bennett, K.A., Vasavada, A.R., Greenhagen, B.T, Schofield, J.T., McCleese, D.J., Foote, M.C., De Jong, E.M., Bills, B.G., Hartford, W., Murray, B.C., Allen, C.C., Snook, K.J., Soderblom, L.A., Calcutt, S., Taylor, F.W., Bowles, N.E., Bandfield, J.L., Elphic, R.C., Ghent, R.R., Glotch, T.D., Wyatt, M.B., Lucey, P.G. (2010b). Diviner Lunar Radiometer Observations of Cold Traps in the Moon’s South Polar Region. *Science*, Vol 330, p479-482. (http://www.diviner.ucla.edu/docs/paige_2010.pdf)

Ramanathan, V. and A. Inamdar (2006). The Radiative Forcing due to Clouds and Water Vapor. In: *Frontiers of Climate Modeling*, J. T. Kiehl and V. Ramanthan, Editors, (Cambridge University Press 2006), pp. 119-151.

Smith, A. 2008. Proof of the atmospheric greenhouse effect. *Atmos. Oceanic Phys.* arXiv:0802.4324v1 [physics.ao-ph] (http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.4324v1.pdf ).

Stephens, G.L., A. Slingo, and M. Webb (1993) On measuring the greenhouse effect of Earth. *NATO ASI Series, Vol. 19*, 395-417.

Trenberth, K.E., J.T. Fasullo, and J. Kiehl (2009). Earth’s global energy budget. *BAMS*, March:311-323

Vasavada, A. R., D. A. Paige and S. E. Wood (1999). Near-surface temperatures on Mercury and the Moon and the stability of polar ice deposits. *Icarus* 141:179–193 (http://www.gps.caltech.edu/classes/ge151/references/vasavada_et_al_1999.pdf)

Thanks for posting this Anthony. The debate on the correct grey-body temperature of the Moon and Earth will be interesting!

Above my paid grade. In fact out of my orbit. But I love it.

Thanks, Anthony, particularly for the caution at the top. I see one typo? in the work:

Seems like part of the sentence is missing.

All the best,

w.

REPLY:Thanks, fixed. Many instances like that where copy/paste from the original document outright fails for some reason. – AnthonyThank you. This will take several readings, and several cups of hot tea to fully digest.

I am trying to get my head around all this, but that’s a good thing. Real science rather than polar Bears fall ing off icebergs. I love it!

I’ll post here what I communicated to Ned Nikolov directly:

Thanks for the reply.

First, I would like to say what I think is good about this reply: It is well-written and clearly explains your thinking and I thank you for that. Furthermore, the actual calculations that you do appear to be correct.

However, I believe there are some quite erroneous statements and interpretations as discussed below that will presumably set the stage for major incorrect conclusions in the second part of the reply.

Here is a brief discussion of the most major errors that I found:

page 3:

This is mainly a semantic issue: The conventional assumption of a constant albedo is used in order to derive the temperature rise attributable to the greenhouse effect alone. It is true that clouds also have another effect, in that they change the planet’s albedo, but that is a separate issue. Furthermore, if you do want to imagine removing the albedo due to clouds, you really also have to then ask how the surface albedo of the colder planet will change due to the increase of snow and ice. (But then…Does the planet still have water on it to form snow and ice? It depends by what magic we got rid of the IR-absorbing gases in the atmosphere! We thus get into various hypotheticals!) I think the best “clean” statement that we have is that if we imagine somehow turning off the radiative greenhouse effect without changing anything else, then our planet would be about 33 K colder.

page 5:

No…You guys have failed to fully understand the implications of Holder’s Inequality. First of all, there is no condition that says that additional thermal energy can’t be added to the planet: It is receiving energy from the sun! Hence, conservation of energy has been misapplied here.

The correct way to apply energy conservation is to note that what is required is that the planet will be near (and always pushed toward) a state of radiative balance where it is radiating back into space the same amount of radiation as it receives from the sun. What Holder’s Inequality shows is that there are lots of different temperature distributions having lots of different average temperatures that satisfy the criterion that the average radiative emission is, say, 240 W/m^2. A planet could thus hypothetically have any average temperature compatible with Holder’s Inequality, which means any average temperature lower than the average temperature for a perfectly uniform temperature distribution (which is ~255 K for a blackbody earth); the actual one that occurs will depend not only on the distribution of insolation (in space and time) but also on the convective, advective, and conductive transport mechanisms in the atmosphere, oceans, and (to a lesser degree) solid surface.

page 9:

It is not really a matter of one being more accurate than another. It depends very sensitively on how you define the average temperature of the surface, e.g., is the surface the first millimeter of the planet’s surface or is it the first meter? The data from Diviner presumably is measuring the temperature right at the surface. However, if you go several centimeters down, you find the temperature remains remarkably uniform between day and night and you get the average value of around 240 K. This is explained, for example, here: http://www.asi.org/adb/m/03/05/average-temperatures.html

Hence, average surface temperature of an airless planet is not a very well-defined quantity because of the large temperature swings right at the surface. This is fundamentally because lots of different average temperatures (corresponding to different temperature distributions) lead to the same amount of total power emitted by the planet’s surface.

page 11:

Not really…What the results, correctly interpreted, show is that it is probably not very useful to talk about an average temperature for an airless planet where the temperature distribution is so non-uniform. That is why scientists usually talk about the “average temperature” determined by averaging T^4 and taking the 4th root.

First of all, you can’t really appeal to intuition here…What is a reasonable thermal enhancement due to the IR-absorbing gases? How is one to know?

More importantly, nobody is claiming that the IR-gases are responsible for such a large thermal enhancement. They are responsible for raising the average emissions of the surface from ~240 W/m^2 to ~390 W/m^2, which for any sort of reasonably uniform temperature distributions (such as that which occurs on the Earth presently) corresponds to a temperature rise of ~33 C.

The limitation imposed by the physics is not directly on the amount of kinetic energy that the lower troposphere can contain. It is receiving huge amounts of energy from the sun! The limit is instead set by radiative balance…i.e., that the Earth must radiate back out into space the same amount as it receives from the sun (or else it will warm up or cool down).

The “among other things” is alas the rub. The Ideal Gas Law alone does not uniquely constrain the surface temperature.

(1) What process is isobaric?

(2) Try as one might, one is not going to explain the fact that the Earth’s surface emits an average of ~390 W/m^2 while the Earth + atmosphere absorb an average of ~240 W/m^2 without acknowledging that the atmosphere must absorb some of the terrestrial radiation emitted, i.e., that there is a radiative greenhouse effect.

Sincerely yours,

Joel

Taking Ned’s ATE as a starting point various implications for climate would follow. In that context I had this exchange with Joel on one of the other threads.

“For a planet without a greenhouse effect, said height is necessarily the surface of the planet”

Irrelevant because energy still gets into the air via conduction and convection and back to the surface via convection and conduction for then radiating out. So there will still be a lapse rate and it must match ATE otherwise the system is unstable. If it doesn’t match ATE then the atmosphere will accumulate energy via conduction and convection indefinitely until it boiled away or lose energy to the ground via conduction and convection indefinitely until it congealed on the surface.

The true Perpetuum Mobile is the concept of a planet with an atmosphere that is not precisely in equilibrium with its ATE.

Any disequilibrium will either boil off the atmosphere or congeal it on the ground.Once congealed on the ground it would be lost via sublimation.

The radiative GHG theory is itself a Perpetuum Mobile because it proposes that changing the composition changes ATE. Thus a bit more human GHGs are amplified by more water vapour and that gives more GHGs which amplifies again ad infinitum.

The atmosphere and oceans would get hotter and hotter until they boiled away.

We would see lots more planet sized bodies with no atmospheres at all because a little change in atmospheric composition would have been enough to destabilise it.

IIf we were to reduce GHGs so that they changed the ATE lapse rate the other way then the Earth would get steadily colder until the oceans and air congealed on the ground.

The system won’t allow it. Any change in composition that might introduce a disequilibrium with ATE is neutralised by a reconfiguring of the circulation pattern.

If you could find one planet where the Gas Laws do not apply then you would have me. Where is it ?”

Willis Eschenbach says:January 22, 2012 at 12:06 pm

Thanks, Anthony, particularly for the caution at the top. I see one typo? in the work:

Comparing the final form of Eq. (5) with Eq. (3) shows that in accordance with Hölder’s inequality.

Seems like part of the sentence is missing.

I notice the sentence following contains the key to understanding the reason for the inclusion of one of the terms which has been referred to as a ‘free parameter’ by some critics.

To make the above expression physically more realistic, we add a small constant W m-2 to So, so that when So = 0.0, Eq. (5) yields Tgb = 2.72K (the irreducible temperature of Deep Space), i.e: eq 6

Willis Eschenbach says:January 22, 2012 at 12:06 pm

Thanks, Anthony, particularly for the caution at the top. I see one typo? in the work:

Comparing the final form of Eq. (5) with Eq. (3) shows that in accordance with Hölder’s inequality.

It’s an error in the formatting. The .pdf reads:

Comparing the final form of Eq. (5) with Eq. (3) shows that Tgb << Te in accordance with Hölder’s inequality.

It is probably useful to go into a little more detail on this claim than I did above by providing a little more discussion as to why Nikolov and Zeller’s claim is wrong here: Let’s suppose they were right, so we take a non-spinning planet and start a planet spinning and the temperature distribution becomes more uniform without changing the average surface temperature. Is this a reasonable steady-state?

Initially, before spinning, the planet was in radiative balance. Now that it has a more uniform temperature distribution but the same average temperature, the average value of T^4 will necessarily be lower. Hence, it will now be emitting less power back into space than it absorbs from the sun. What we such a planet do? It will warm up until such point as it is now emitting the same amount of power as it is absorbing from the sun. Hence we see how energy conservation, correctly applied, constrains not the average temperature but the emitted power.

In principle it looks fine but relys heavily on the grey body definitions and calculations, obviously. Quite a few typos and missing words I think but so what. I know how difficult ( and time consumming) that would have been for you Anthhony.

“Try as one might, one is not going to explain the fact that the Earth’s surface emits an average of ~390 W/m^2 while the Earth + atmosphere absorb an average of ~240 W/m^2 without acknowledging that the atmosphere must absorb some of the terrestrial radiation emitted, i.e., that there is a radiative greenhouse effect.”

The surface of the Earth exchanges 150W/m2 by a dynamic conductive exchange between surface and atmosphere.

The Earth also exchanges 240W/m2 with the incoming solar energy from the top of the atmosphere.

Joel says the atmosphere gets its energy solely from downward radiation from GHGs. The truth is that it is conduction and convection.

Wiki is quite clear on the issue:

http://en.wikipedia.org/wiki/Lapse_rate

“Because the atmosphere is warmed by conduction from Earth’s surface, this lapse or reduction in temperature normal with increasing distance from the conductive source.”

And it all applies to:

“any gravitationally supported ball of gas.”

without any breach of the Laws of Thermodynamics.

With respect, Ned and Karl, didn’t NASA ask-and-answer this question back in the 1960s?

NASA asked the question in an engineering context: “Can a huge (> 80%) improvement in thermal insulation [in a rocket stage] be the result of a few grams of reflective plastic foil, that collectively amount to less than 0.0001% of total cryogenic booster mass? …”

NASA’s answer (which obvious in retrospect) was “tiny thickness of foil create a huge increase in insulation.”

That is why (it seems to me) that your entire last paragraph should be dropped, for the simple reason that engineers familiar with radiative heat transfer will immediately appreciate that its reasoning is just plain wrong.

We have over 260 comments on the thread we have been running at the Talkshop about this paper since Jan 17th. It’s a hot topic!

Ned (or anyone), I don’t understand this claim:

The mean equilibrium temperature, as Holder’s inequality establishes, cannot exceed what you are calling the “effective emission temperature” (Equation 3).

However, for a

given input of solar energy, the mean equilibrium planetary temperature can take up a variety of values as long as it is less than the effective emission temperature. For a planet the value is mainly set by the length of the day (speed of rotation) and the presence/absence of an atmosphere.Again for a

given input of solar energy, the rule is “the bigger the temperature swings, the lower the temperature average”.So we can have two identical planets whose only difference is rotation speed. Both are receiving the identical amount of solar energy. The planet that is rotating faster will have a higher mean equilibrium temperature without any “net input of extra energy”.

Which seems to contradict your statement that I quoted. What am I missing?

All the best,

w.

Personally, I don’t see anything wrong in their explanation (a couple of minor typos though) as a basic description of an assessment of direct evidence (measurement) of a comparable irradiated object (the Moon) and its theoretical SB derived temperature. As I see it, they have then ‘reverse engineered’ to dare to suggest that the SB derived temperature for the earth (and the subsequent assumed GHE) is potentially in error.

In my opinion, there is definate merit in detailed independent analysis – especially concerning the math around Holders inequality (!). The thinking is indeed somewhat ‘out of the box’ – but it should not be discounted ‘out of hand’ and if the actual measurements from the moon are not meeting ‘theoretical’ expectation they are absolutely correct in raising this…their actual theory may be somewhat wrong or overstated, but the big question, as I see it – is that the accepted theory (SB derived temp) may well not be correct either!

Very elegant and convincing. I keep looking for some huge hole in the logic (it must be there – mustn’t it?). It will be a tough row to hoe because it makes most of the climate community look so foolish. Why do we so easily and often get the cart in front of the horse??

Right or wrong, this will be one interesting subject to follow, in the coming future. Hopefully it will shake out new ideas and approaches. GK

tallbloke says:

Nope…Except at the very earliest stages of comments of their original paper, I have not been counting this value as a free parameter because I have recognized that it is not really free. The four parameters that are really free are those in Equation (7) of their original paper.

Nice presentation, except that it ignores conduction transfer of energy from earth’s surface to the atmosphere.

Imagine a ball of nitrogen & oxygen gas, bound up in a zero thickness, purely transparent shell, in deep space. With the “local environment” existing at barely warmer than absolute zero, it is reasonable to assume that our ball of non-greenhouse gases would obtain a comparable temperature.

Now, move that ball to the vicinity of the moon, with the sun’s radiation pouring into it. Since the gases are nearly (not quite 100%) transparent to the sun’s range of emitted radiation, as well as to the moon’s re-radiation wavelengths, our nitrogen & oxygen still would remain only slightly warmer than in deep space.

If instead, our ball were filled with CO2 and other “greenhouse gases,” then our new ball would warm up significantly, since those gases do indeed absorb the moon’s re-radiation.

This is a distinction which exists only in outer space, and this usually is used to justify the distinction here on earth. I believe this is a flawed interpretation of how the system works. Our environment here on earth is quite different from outer space. Here, other forms of energy transfer are at play – conduction and convection.

Let us imagine once more. This time, we have an earthlike planet, covered by an atmosphere made up entirely of nitrogen and oxygen – with no so-called greenhouse gases. As in deep space, our sun’s radiation essentially passes through our atmosphere, with negligible absorption.

When that radiation reaches earth’s surface, some of that energy is reflected, while most is absorbed by the land and seas. (So long as we’re imagining, let’s assume water that doesn’t evaporate into the atmosphere.)

This incident radiation striking the earth warms up the surface. While some of this heat is reradiated back into space, once again passing through our IR transparent gases, much of the heat is conducted to the atmosphere. (This article admits that, on the moon, “Surface temperatures at low latitudes soar to 390K (+117C) around noon…”)

The air that is touching the earth’s surface, warms, becomes less dense than the cooler air above it, and rises, leaving cooler air now in contact with the surface. Convection currents work even if there are no greenhouse gases.

Since the sun beats down on only half our earthly disk at a time, the sun facing side warms, through radiation from the sun, compared to the dark side. Planetary differences between surface temperatures cause planet scale convection patterns in our nitrogen-oxygen atmosphere.

These convection patterns will warm even the night time earth above a bare earth temperature. Yes, this nitrogen-oxygen atmosphere will lose energy radiation to space, but it also will radiate partly back onto the earth from which it originally received its energy through conduction. Though the nitrogen and oxygen have warmed through conduction rather than radiation absorption, from that point forward they essentially are greenhouse gases.

Our “non-greenhouse gas” earth system still will warm considerably compared to an atmosphere free planet, and this warming will be well above the SB black body equilibrium of an airless body in space. How much warming our earth would experience, I will leave for the physicists to determine, but ignoring conduction and convection energy transfer mechanisms is not acceptable.

One consequence of our considering only radiative absorption as the way our atmosphere can warm, is that we ascribe the entire difference in earth’s temperature, between its current level and what it would be without any air at all, to greenhouse gases. We are conceding ground to the CAGW activists by default.

Our non-greenhouse gas system still acts as a greenhouse; our non-greenhouse gases indeed are greenhouse gases. The whole distinction based on how the various gases receive their energy, some of which then is radiated partly out to space and partly back to earth, is false.

We cannot determine how much CO2 and the IR absorbing gases raise earth’s temperature without including conduction absorbing gases.

Willis says:For a planet the value is mainly set by the length of the day (speed of rotation) and the presence/absence of an atmosphere.

Again for a given input of solar energy, the rule is “the bigger the temperature swings, the lower the temperature average”.

Sounds about right to me. So the Moon is likely considerably colder on average than the long accepted calcs say it is. But since Earth spins more rapidly (once a day rather than once a month) then even without an atmosphere it would be warmer. It’s axial tilt has to be accounted for too.

Stephen Wilde says:

No…You are attributing things to me that I have not said. What I have said is that the fact that conduction and convection exists doesn’t get you around the fact that the Earth with its current surface temperature is emitting ~390 W/m^2 and hence that, in the absence of an IR-absorbing atmosphere, the Earth system would be emitting ~150 W/m^2 more than it is absorbed from the sun and would rapidly cool down.

Formatting problem in Table 1: last column should read 143 instead of 14 for the south pole ATE. The cursor seems to hide the last figure…

Very interesting discussion! (No fishing lines this time… 😉 )

Consider two bodies, A is fast rotating and therefore has a near uniform temperature, and B is rotating slowly and consequently has huge temperature differences.

From the non-linearity of eq. (1) you can see that if A and B have the same mean temperature then B will emit more radiation than A. To understand that, you can for example say that A is uniformly at 300K and B has one half at 200K and the other at 400K, and do the math.

If A and B are in the same distance from the sun and have the same albedo, they will receive the same incoming radiation.

Remember that the outgoing radiation has to equal incoming radiation over time, ie. the outgoing radiation from A has to equal the outgoing radiation from B. Therefore the mean temperature of A has to be higher than B. This proves that equation (6) is wrong for a rotating body.

Mr Nikolov, I will sugest to read this paper by acad. Sorohtin:

http://fiz.1september.ru/articlef.php?ID=200501111

This is in Russian but propbably you will not have problems. I think this is quite close to your theory.

// sarc on //

God, I hate this stupid denier site. No science ever goes on here!

// sarc off //

Great job here. It’s above my pay grade, as they say, but I’m learning tons.

This on the surface at least looks to me a systematic approach to the problem that is at least accessible to the lay person.

errata: just before equation 2 should it not be So not Sa for TOA solar irradiance?

It raises some questions that I have had about the orthodox approach which may already be covered in the parts I haven’t yet read… will ask them later if not.

There is a big problem in the integral when they make the substitution u=cos(theta).

if u=cos(theta) you have to express dtheta using du to make the substitution.

dtheta=-du/sqrt(1-u2)

So after substitution you have something with u^0.25/sqrt(1-u^2) to integrate, and certainly not u^0.25 only!!!

http://www.sosmath.com/calculus/integration/substitution/substitution.html

Sorry, but this is plain terrible.

Roger says:

Sounds about right to me. So the Moon is likely considerably colder on average than the long accepted calcs say it is. But since Earth spins more rapidly (once a day rather than once a month) then even without an atmosphere it would be warmer. It’s axial tilt has to be accounted for too.

I don’t follow how rotation/spinning increases temp – logically (and ignoring any internal heat generation etc) the sum of the irradiance must equal the sum of the re-radiance i.e. energy in = energy out ? – in the no atmosphere scenario, why would spinning make the temp warmer? The diurnal change (or whatever frequency of change one has!) doesn’t change the total amount of solar radiation received nor the total amount of reradiatione emitted!

In summary, may we conclude that circulating atmospheric pressure is a hitherto neglected factor that in proper SB context both empirically and theoretically multiplies Earth’s previously hypothesized global equilibrium temperature by ~1.86, nearly doubling previous estimates?

On this basis, given requisite variations on such an enhanced figure, it seems that Gaia’s standard range of temperature fluctuations encompasses virtually all geohistorical extremes from steaming Jurassic jungles to pre-Cambrian ice ages. This suggests that not global atmospheric/oceanic circulation but plate-tectonic dispositions have been the root cause of Earth’s non-cyclic but periodic warming-cooling phases from the planet’s earliest times.

As geophysical phenomena, Pleistocene Era glaciations averaging 102,000 years interspersed with interglacial epochs of median 12,250 years, are accordingly symptoms of a more fundamental “fluid dynamic,” that is, of Earth’s essential thermodynamic planetary composition/structure. If so, any and all AGW hypotheses are iatrogenic artifacts not of empirical investigation but of scholastic preconceptions– not objective, rational conclusions but mere semantic exercises, words.

“Note that, we are not presently concerned with the composition or infrared opacity of the atmosphere. Instead, we are simply trying to quantify the overall effect of our atmosphere on the surface thermal environment”

This includes ‘air’ (atmosphere) pressure?

As complex as this is, it seems to me to be missing something.

If the air at the surface is still (no convection), and the resulting surface temp is T0, then what happens when we add wind (i.e. convection)?

The windy air will convect energy away from the surface, so less energy can be radiated (from the surface), and its temperature must be T1 < T0.

To me is seems implausible that all else being equal, the surface temperature of a planet with little near surface atmospheric turbulence could be the same as that of a planet with lots of turbulence.

the difference in temperature between the moon and the Earth cannot all be down to temperature swings and length of day ,the difference in temperature is still there when we compare the poles of the moon with the poles of the earth.I think it is unreasonable that our atmosphere does not insulate us from the cold of space not just ghg.I think thermodynamic equilibrium is an unreasonable assumption if the Earth and the moon are warmer than space then heat will flow from the moon and the Earth to space .

I am not surprised that Anthony does not support this paper given the prominance and support he gave to Willis Eschenbach’s outburst. But to deny the laws of physics seems a daft tad to me.

Joel Shore says

” I think the best “clean” statement that we have is that if we imagine somehow turning off the radiative greenhouse effect without changing anything else, then our planet would be about 33 K colder.”

Notice the without changing anything else logic.

Here’s Joel’s logic applied to a typical small car

Average speed = 45miles per hour

Fuel consumption in a year 5000 litres.

The engine is now removed without changing anything else.

Fuel consumption drops to zero, excellent carbon footprint numbers, average speed slightly increased because of reduced weight.

Ned (or anyone) I think there is an error between equation 4 and equation 5.

In equation 4 you have correctly indicated that when the sun is below the horizon, the value of Ti is zero.

However, in equation 5 you have only integrated that over half of the surface of the sphere, the sunlit half. This is indicated by mu (the cosine of the zenith angle) varying from 0 to 1, rather than -1 to 1.

Integrating over the full surface gives a different answer than you have. Here’s Mathematica on the subject:

Note that this is just 1/2 of your answer, as we would expect when we include the unlit side.

In addition, to do the integral, in the derivation of equation 5 you have substituted

mufor thecosine(theta)term in Equation 4. I’m not sure you can do that when you are going to integrate. My suspicion is supported by the fact that integrating over cos(theta) from -Pi/2 to Pi/2 gives a very different answer involving the Euler gamma function. Mathematica again:This gives a value of 0.337 in place of the “2/5” at the left of your answer.

Comments?

w.

[EDITED TO ADD: I have corrected an error, I’d incorrectly integrated from -Pi to Pi, it is from -Pi/2 to Pi/2]

The integral of u^0.25/sqrt(1-u^2) from 0 to 1 gives a little bit less than 0.5 (quick numerical approximation).

So what you finally get with a correct substitution (and dividing the integral by 2PI the surface of half a sphere as the integral is made on half a sphere!) is not Tgb=2/5 […]^0.25 but Tgb~1/2[…]^0.25 that is to say 1.6 times your result.

For your example with earth it gives Tgb~1.6*154.7=247.52 K

“”””” where ϵ is the object’s thermal emissivity/absorptivity (0 ≤ ϵ ≤ 1 ), and σ = 5.6704×10-8 W m-2 K-4 is the SB constant. A theoretical blackbody has ϵ = 1.0, while real solid objects such as rocks usually have ϵ ≈ 0.95. In principle, Eq. (1) allows for an accurate calculation of an object’s equilibrium temperature given the amount of absorbed radiation by the object, i.e “””””

This is a rather gross simplification and leads to erroneous calculations. For a start, there is ne spectral emissivity data, and published spectral emissivity data for many types of common rocks give quite different numbers. It is true that many common rocks have an emissivity around 95% at wavelengths longer than about 12 microns, and also below 8 microns, but in that crucial window from 8 to 11 microns, common rocks have considerably lower emissivities, often below 80%.

And this region; 8-12 microns is just the range where the 288 mean earth LWIR radiation spectrum peaks. So the peak emittance from most rocks is omewhat less than the above would suggest, and this is also where the atmospheric window is that allows much of this radiation to escape.

As for “Holder’s Inequality” (with an umlaut), is this some peer reviewed Nobel Prize winning fundamental discovery of Physics; or is it some self evident high school mathematical conclusion. I submit it is “climate scientists” trying to sound important by attaching some glorious title to trivial mathematics.

And any hobbyist playing around with simple elctronics; Radio Shack style is well aware of the fact the RMS value of a sinusoidal waveform is quite different from the average value (which is zero), and also from the average value of the rectified sine waveform. So stop trying to make a big deal out of trivia. If it was Eric Holder, the current Attorney General of the USA, it would make more sense; it’s about on par with his intelligence.

By the way; the very same reference text book, where rock emissivity measured data can be found, also shows BLACK BODY LIKE SPECTRA for the sky radiation as seen from the earth surface; and that spectrum (observed) has the shape expected for the ATMOSPHERIC TEMPERATURE. This despite the well known FACT among climate scientists that ordinary diatomic or monoatomic neutral gases do not emit thermal radiation, as do all other materials at Temperatures abou zero Kelvins.

I see a lot of thrashing around in this paper, including some interesting stuff, but I think it is a bit self congratulatory to describe it as “a unified theory of climate”. In what way is it “unified” ?

“I think the best “clean” statement that we have is that if we imagine somehow turning off the radiative greenhouse effect without changing anything else, then our planet would be about 33 K colder.”

Clean? Clean in science is no missing stuff, not painting over the smudges.

Also in the above analysis of this “unified theory” where is the part about atmospheric H2O and CO2 absorbing part of the incoming solar radiation spectrum, as between 0.7 and 4.0 microns; which itself will warm the atmosphere. Both CO2 and H2O have overlapping (but not coincident) IR bands at 2.7 microns, where a significant solar energy component still exists.

Just what is it that you suppose converts an extra-terrestrial 1362 W/m^2, into a ground level number more like 962 W/m^2

Yes there are two big big errors in this integral:

1) they divide the result by 4pi which is the surface of the entire sphere but the integral is made on half the sphere (cos theta varying from 1 to -1 covers the north hemisphere)

2) the substitution u=cos(theta) is a big big failure…it does not give u^0.25du but it gives

u^0.25/((1-u^2)^0.5)du

And the final, correct result is approximately 1.6 times the result presented here->247K. Is it some pro AGW people trying to show we are stupid to accept terrible mistakes if they tend to show things differently ??

@Joel Shore:

“I think the best “clean” statement that we have is that if we imagine somehow turning off the radiative greenhouse effect without changing anything else, then our planet would be about 33 K colder.”

But this is an issue they are disputing, and it is a key point they are making. It is almost as you are saying “I do not want to look at step F, I’m contempt with things as they are” while their argument is step A, step B, step C etc. through step F all the way to a final conclusion. It is a faulty dismissal.

I’m inclined to think they have missed something obvious in all their steps, but I still accept that they must be allowed to have their logic heard.

And also, I for one think it is entirely fair to set up a Case Scenario where earth has no atmosphere and albedo at 0.12. It is the current scenario they are discussing. Not yesterday or tomorrows scenario with more or less snow, or whatever.

Bob Shapiro says:

January 22, 2012 at 1:22 pm

Our non-greenhouse gas system still acts as a greenhouse; our non-greenhouse gases indeed are greenhouse gases. The whole distinction based on how the various gases receive their energy, some of which then is radiated partly out to space and partly back to earth, is false.

I’ve written an analysis to that effect. Reviewers welcome.

http://tallbloke.wordpress.com/2012/01/20/greg-elliott-use-of-flow-diagrams-in-understanding-energy-balance/

The paper also identifies the assumption of constant albedo as a source of error in climate models.

The entire conclusion of Nikolov and Zeller rests on the assumption that the Moon is a gray-body equivalent to the Earth, and this assumption is quite erroneous. The Earth, because of having a far more dense and thermally active interior, generates far more longwave radiation than the Moon, leading to of course the fact that the surface of the Earth emits more net energy than it absorbs through sunlight. This energy of course leaves the ground and is abosrbed and re-emitted by the atmosphere in proportion to the concentration of GH gases, which leads to higher lor lower atmospheric temperatures, depending of concentration levels. The geologically dead and rather inert Moon on the other hand emits a tiny amount of LW naturally, beyond the solar energy absorbed, but it is at a tiny faction of what the Earth naturally emits beyond solar energy absorbed. This greater amount of energy generated by the surface of the Earth versus the Moon is accounted for in Trenberth’s energy balance etimates in upward directed LW, and leads to the fact that the conclusion by Nikolov and Zeller, that the ATE is ‘3 to 7 times higher’ than currently estimated, is also quite erroneous. In short, because of the very geologically and thermally active nature of the Earth (the core could be as high as 5000C or so), the heat generated by the surface of the Earth itself is not inconsequential, and makes the rather inert and thermally dead Moon a very poor and highly inaccurate grey-body equivalent to the Earth.

“I see a lot of thrashing around in this paper, including some interesting stuff, but I think it is a bit self congratulatory to describe it as “a unified theory of climate”. In what way is it “unified” ?”

Considering the average dogma indoctrination rate of climate scientists its probably more than worthwhile to point out its far from divinely-inspired perfection.

Since there is no such a thing as “global annual temperature”, but rather huge number of local temperature systems, this is all lot of theoretical arguments about some virtual planet that does not exist! Please comment on Essex et al 2007, Kramm and Dlugi 2011 and Pell et al (2007) on Koppen-Geiger climate classification in Pell et al 2007.

DB-uk

“According to Eq. (2), our atmosphere boosts Earth’s surface temperature not by 18K—33K as currently assumed, but by 133K! This raises the question: Can a handful of trace gases which amount to less than 0.5% of atmospheric mass trap enough radiant heat to cause such a huge thermal enhancement at the surface? Thermodynamics tells us that this not possible.”This is really nuts, and is being repeated. No-one claims that the atmosphere boosts surface temperature by 18K-33K. Some say that the GHG fraction of the atmosphere has that effect, relative to an atmosphere with no GHG.

But to then say that the 133K is due to a handful of trace gases when you’ve removed the whole atmosphere????

The GHE is that difference between air with and without GHG. Your ATE is something else.

I cannot accept your reasoning for being allowed to ignore rotation of a planetary body.

Imagine an infinitely fast rotating body. It would be the equivalent of incoming radiation falling on the whole surface of the planet, and not just half of it. This would result in the “shaded” side having a higher, and the “lit” side a lower temperature. And because of the 4th power law and Hölder’s inequality you would get a higher average temperature.If your further conclusions rest on this averaging, they cannot be trusted.

Proof me wrong by doing the integration over time, or show explicitely the “fallacy” of doing it.

Willis Eschenbach says:

No, Willis. They have just used the fact that the integral of an integrand that is zero (as it is over the dark half) is zero. I think for the approximation that they are making (i.e., that the local temperature is determined by radiative balance with the local insolation), their calculation is correct.

Ned & Karl:

I do see one possible slip in the text. It states:

“To make the above expression physically more realistic, we add a small constant Cs =0.0001325 W m-2 to So, so that when So = 0.0, Eq. (5) yields Tgb = 2.72K (the irreducible temperature of Deep Space), i.e: …”; but 0.0001325 is closer to 5 K than 2.72 K stated. Maybe the text should be altered to state that 5.07K is the CMB plus mean star-shine that all interstellar bodies would also receive at all times. That is what I assumed that tiny difference represented.Anthony Watts.

Having just read the foreword, I paused before getting in to the main post because the only thing going through my mind was “what a gentleman Anthony is.

He is obviously hard working and intelligent, but he is also polite, courteous and considerate of others, hence the foreword.”

Anthony, you’re a credit to yourself and your family. I’d like to meet you one day, maybe the next time you come to Australia. I’d like to shake your hand.