Earth's Energy Flows

by Miklos Zagoni

Eotvos Lorand University
Budapest, Hungary

miklos.zagoni@earthenergyflows.com

We found four equations for the Earth's annual global mean energy flow system — two for the clear-sky, two for the all-sky; one pair for the net radiation at the surface and one pair for the total (absorbed and emitted) energy at the surface —, coupling surface fluxes directly to top-of-atmosphere (TOA) fluxes, namely, to outgoing longwave radiation (OLR) and the longwave cloud radiative effect (LWCRE). On NASA CERES satellite-based observational datasets (EBAF Edition 4.1, Version 3, 22-year global means from April 2000 through March 2022), the individual bias of the equations is less than ±3 Wm-2 and the mean bias of the four equations is 0.0007 Wm-2.

On this page I will demonstrate that these four equations are immediate consequences of the simplest greenhouse geometry as described, for example, in Dr. Kevin E. Trenberth's recent book, "The Changing Flow of Energy Through the Climate System" (Cambridge University Press, 2022).
• I will show that the fundamental equations given by Dr. Trenberth to describe his model are directly related to Karl Schwarzschild's (1906) two-stream radiative transfer equation.

• I will point out that the geometric model represents the simplest arithmetic ratios of 1 : 2 : 3 : 4, or their extended version, 1 : 2 : 3 : 4 : 6 : 8. I will prove on published datasets that the Earth's global mean clear-sky energy flow system accurately satisfies these ratios.

• I will prove that this clear-sky integer system can be extended to the all-sky in one single step. By using the longwave cloud radiative effect (LWCRE at TOA) as one unit, the all-sky energy flow system also reveals a simple integer ratio structure, with small integers as multiples of LWCRE. This system satisfies the four equations far within observational uncertainty.

I will demonstrate that the first of these four equations comes from Schwarzschild's original (1906) two-stream radiative transfer equation and has been consistently reproduced in standard university textbooks on atmospheric physics and radiation — from Goody (1964, 1989, Oxford) to Houghton (1977, 1986, 2002, Cambridge) and Ambaum (2021, RMetSoc) — but is notably absent from all the climate change literature, including studies and assessments from Manabe-Strickler (1964) and Manabe-Wetherald (1967) to the IPCC reports (1990–2021).
  • I will show that the integer systems are accurately verified on published energy budgets, including the 30+ years of GEWEX datasets, 24 years of NASA CERES satellite observations, and two decades of CMIP climate models, cited in IPCC reports.
  • To illustrate the accuracy: the clear-sky normalized greenhouse factor g(clear) has the integer position of 1/3 in the geometric structure; its value on the CERES EBAF Ed4.2 (24-yr mean) is 0.33313. Its all-sky equivalent is g(all) = 2/5, while the IPCC AR6 has 0.3995. The two all-sky equations fit into their integer positions with a difference of 0.1 Wm-2 on the GEWEX dataset (Stephens, 2023, BAMS) and, with a difference of 0.6 and 0.08 Wm-2, resp., on 22 years of CIMP6 data (Wild, 2024, Nature Comm.). 
  • A highly counter-intuitive recognition is that solar reflections, both clear-sky and all-sky, precisely occupy integer positions within the global mean energy flow system without referring to any atmospheric, cloudy or surface reflective property—an observation pointing far beyond the ruling paradigm and allowing Total Solar Irradiance (TSI) to include in the integer ratio system with a definite position.
  • The aim of the recent work is to make public these recognized relationships and to offer a possible theoretical explanation. I do not want to go into speculations on how the channels of human influence on the climatic system should be reconsidered in light of these new findings.
It seems the prevailing theory has at least two critical shortcomings: it fails to account for the integer structure within the global mean energy flow system, and it overlooks a long-established radiative transfer constraint equation found in textbooks—either of which is fatal on its own. At this stage of my research, I do not claim to present a complete alternative theory. However, I assert that these two essential aspects must be addressed in any robust understanding of our climate. 

The historical development of my observations followed a reverse order. I first identified integer ratio systems in published global mean all-sky energy budget assessments (e.g., Stevens and Schwarz 2012; Stephens et al. 2012; Wild et al. 2013; Loeb, 2014), then in the clear-sky CERES EBAF (2016) datasets. I then located the corresponding equations in the history of radiative transfer (e.g., Schwarzschild, 1906) and validated the system using recent data from GEWEX (2023), NASA CERES (2024), and CMIP6 (2024). Encountering Dr. Trenberth's greenhouse geometry in his recent book was a significant and confirming moment.
II.  PRESENTATIONS

HISTORICAL INTRODUCTION (Max Planck Institute, Hamburg)
Observations reveal integer ratios between the components of Earth's global mean energy flow system
Observing and Modeling Earth's Energy Flows — Twelve Years Later (video, 39:46)

THEORY (THE PHYSICAL SCIENCE BASIS)
The integer ratios are solution of four basic radiation transfer equations
Trenberth's Greenhouse Geometry I.(video, 1:16:09)

IMPLICATIONS, INTERPRETATIONS, SPECULATIONS
The validity of the equations has far-reaching consequences
Trenberth's Greenhouse Geometry II,: Implications (video, 1:11:21)

AMS 2025 ANNUAL MEETING (New Orleans, Poster 1)
The clear-sky equations are equivalent to Dr Trenberth's Earth's-like geometric greenhouse model
Trenberth's Greenhouse Geometry and its Representation on the Earth's Atmosphere (Poster)

AMS 2025 ANNUAL MEETING (New Orleans, Poster 2)
There is a long-known equation constraining convection 
A Constraint on Convection (Poster)

AGU 2024 FALL MEETING (Washington, iPoster 1)
Gewex data verify the all-sky equations by very high accuracy
Modeling and Observing Global Energy and Water Cycles by GEWEX (iPoster)

AGU 2024 FALL MEETING (Washington, iPoster 2)
A valid radiative tranfer constraint equation on convection is omitted from climate models
What's Next for Science: Theoretical Understanding of Atmospheric Convection on Global Scales (iPoster)

EGU 2024 GENERAL ASSEMBLY (Vienna, talk)
Dr Trenberth's greenhouse model is an accurate description of Earth's global mean clear-sky energy flow system
Trenberth's (2022) Greenhouse Geometry (EGU talk)

NASA CERES SCIENCE TEAM MEETING (Washington, talk)
The fundamental integer relationships presented to NASA in 2017
Patterns in the CERES Global Mean Data (2017 NASA GSFC)

EARTH RADIATION BUDGET WORKSHOP / NASA CERES-Libera Science Team Meeting talk (Hamburg, pdf)
Updated CERES observations verify the equations and their integer solution accurately
Observing and Modeling Earth's Energy Flows, Ten Years Later (Max Planck Institute, Hamburg, Germany, 2022) (pdf)

AGU 2021 Fall Meeting iPoster
https://agu2021fallmeeting-agu.ipostersessions.com/default.aspx?s=BC-5C-80-85-7C-F7-AF-C1-6C-12-83-4B-1D-AC-46-AA&guestview=true

AGU 2020 Fall Meeting iPoster
https://agu2020fallmeeting-agu.ipostersessions.com/default.aspx?s=BD-5D-34-45-A3-93-F7-C1-40-CA-81-80-4F-97-74-E1&guestview=true

AMS 2021 Annual Meeting
https://ams.confex.com/ams/101ANNUAL/meetingapp.cgi/Paper/376937

EGU 2021 General Assemby
https://earthenergyflows.com/Sir_John_Houghton_on_radiation_transfer-by-Miklos_Zagoni_v720.mp4


III. HISTORY

My first encounter with the integer system: Stephens et al. (2012, Nature Geosci.) (video, 4 min)


* * *
The second, next year: Wild et al. (2013, Clim.Dyn., IPCC AR5) (video, 3 min)


* * *
The third, next year: Loeb (2014, NASA CERES) (video, 1 min)

IV. THEORY, PART 1. TRENBERTH'S GREENHOUSE GEOMETRY
(video, 18:32)




THEORY, PART 2. EQUATIONS
(video, 20 min)


THEORY, PART 3. GEOMETRIC DEDUCTION
(video, 9 min)

For more of the theory, see the full video:
Trenberth's Greenhouse Geometry, The Physical Science Basis


V. BOTTOM LINE / TAKE AWAY

Stephens et al. (2023, GEWEX, BAMS), Wild et al. (2024 Nature Comm.,), CERES (2024) (video, 2 min 20 sec)


* * *
VI. A note on Total Solar Irradiance (as the basis of the unit flux):


VII. PLAIN LANGUAGE SUMMARY

If you are familiar with the global warming literature, you know that, according to the ruling theory, more CO2 in the atmosphere is thought to increase the greenhouse effect and modify other components in the global mean energy flow system. Now a very simple observation shows that in the published reliable global energy flow estimates there are given ratios, expressed in small integer quotients, between these energy flow components. They seem to behave as huge ‘quanta’, where the prescribed value, for example, for the all-sky greenhouse effect as 6/15, that is, 2/5 = 0.4, cannot be modified slightly to a higher value of, say, 0.43, by greenhouse gas emissions. This is a very unexpected feature, but as every flux components in the annual global mean energy flow system (both in the clear-sky and all-sky, at the top of the atmosphere, within the atmosphere and at the surface) are consistent to the arithmetic ratio system within observational uncertainty, I decided to take it seriously for a moment and started to investigate it in detail.

In this webpage you can see what I found. I point out the integer systems in several published global energy budgets, I demonstrate that the numbers in these integer systems are solutions of very simple and well-known radiation transfer equations (some of them are taught in standard university textbooks on atmospheric physics and radiation), and I deduce the system of these energy flows from the simplest greenhouse models like the one described in Dr Trenberth’s most recent book. — After studying these structures for more than a decade, I became familiar with the equations and their integer solution, but I must confess: the accurate fit of reflected solar radiation to integer positions at the top of the atmosphere, both in clear-sky and all-sky, is still beyond my comprehension.

Although the goal of this study is to present these observations to the Reader without going into speculations on how and why our atmosphere is able to maintain these steady-state quantal structures, or where might be the limits of this stability, in the 'Implications' video I still try to give an explanation (via an equation) for the possible reason of this geometric behavior. It is evident that our system is warming, and it is also understood that, in recent decades, the primary cause of this warming is the increase of absorbed radiation from the Sun. But to investigate the possible channels of human perturbations on the system would lie far beyond the scope of my recent efforts. I only wanted to emphasize that the greenhouse factors (both in clear-sky and all-sky) seem to sit very accurately on their prescribed theoretical GHG-independent geometric positions.

  


* * *
VIII. SUMMARY FOR POLICYMAKERS

The failure of IPCC to account the recent solar-absorption induced global warming, to predict the constancy of the greenhouse factor, to reproduce basic theoretical equations and to reflect the apparent flux integer ratio system is a consequence of their over-confidence. They might have inherited it from the Charney Report (1979): “we have tried but have been unable to find any overlooked or underestimated physical effects that could reduce the currently estimated global warmings due to a doubling of atmospheric CO2”. Using the “virtually certain” and “very high confidence” language, but being wrong in them, they fall because they never added: according to our recent knowledge. Jule Charney was an excellent scientist working with ENIAC on numerical weather prediction, but with no experience in radiation transfer theory. They completely overlooked the constraint equation on convection, although it was available in standard university textbooks on atmospheric physics and radiation in that time (Richard Goody: Atmospheric radiation: Theoretical basis, Oxford University Press, 1964; John Houghton: The physics of atmospheres, Cambridge University Press, 1977; Joseph Chamberlain: Theory of planetary atmospheres, Academic Press, 1978), and was there in university lecture notes and even in classroom exam excercises. You cannot have an A if you are unaware of it! Without that constraint, any assessment of the physical science basis of our climate is incorrect. — How could it happen that an entire profession ignored it for 30+ years? The only excuse could be the lack of reliable global data to control it, almost until recently. The omission of this equation made them blind to other, similarly important fundamental facts: the apparent integer structure in the global mean energy flow system and the corresponding set of governing equations. The observed values sit very accurately in their prescribed theoretical greenhouse-gas independent geometric positions, including the greenhouse factors as well. — But the fact that the ruling theory is wrong does not mean that there is no human influence on the climate system. There is an eminent decrease in solar reflection, leading to robust warming in recent decades. To this day, it is not fully understood whether this is a consequence of aerosol reductions, or an indirect effect of our GHG-emissions? What does seem evident, according to our recent knowledge, that this influence is not via the frequently emphasized greenhouse effect. —  Study science, but follow the precautionary principle: DON'T RELEASE GASES INTO THE ATMOSPHERE! 

CERES EBAF Edition 4.2 Version  4 (released 11/25/2024)
24 running years 288 montly global means (April 2000 - March 2024)
(last 24 months are displayed only)


Largest deviation from the integer position at TOA = 1.52 Wm-2 (SW CRE); at the surface = 3.22 Wm-2 (SW up all-sky)

g(clear-sky, theory) = (15 – 10)/15 = 1/3,   g(clear-sky, CERES) = (398.77 – 265.97)/398.77 = 0.333
g(all-sky, theory)    = (15 – 9)/15 = 2/5 = 0.4, g(all-sky, CERES) = (398.62 – 240.39)/398.62 = 0.397

Download data in MS Excel file:
 https://earthenergyflows.com/04-2000-03-2024.xlsx

References

Schwarzschild, K. (1906) Ueber das Gleichgewicht der Sonnenatmosphäre. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse.

The original two-stream approximation of the radiation transfer problem in a gray atmosphere in local thermodynamic equilibrium. Equation (11) gives the emission of a layer E (for example, the surface), the upward beam A and the downward beam B as a function of the emerging flux at the top-of-atmosphere A0 and the gray ‘optische Masse’ (optical depth) m. — The difference of the upward beam and the emission of the layer gives a net radiation which is constant (independent of the optical depth) and equals half of the emerging flux.

Emden, R. (1913) Über Strahlungsgleichgewicht und atmosphärische Strahlung. Sitzungsberichte der mathematisch-physikalischen Klasse der Königlich Bayerischen Akademie der Wissenschaften zu München. English translation: Radiation equilibrium and atmospheric radiation, by H. Bateman. Monthly Weather Review (1916).  

Emden realized that in Schwarzschild’s (1906, Eq. 11) there is a discontinuity of the Planck-function at the surface in radiative equilibrium, requiring a temperature discontinuity at the surface in radiative equilibrium; but within the same sentence he is added that this “jump” (Temperatursprung) is greatly diminished by evaporation of water and conduction; this way establishing the concept of radiative-convective equilibrium. The paper computes correctly the size (magnitude) of the temperature jump (20°C), but does not display explicitly the constraint on it (A0/2).

 Schwarzschild, K. (1914) Über Diffusion und Absorption in der Sonnenatmosphäre. Sitzungsberichte der Königlichen Preussischen Akademie der Wissenschaften, Berlin. Vorgelegt von Hrn. Einstein.   

This paper gives the general solution of the radiative transfer problem in the form of differential equations.

Milne, E.A. (1930) Thermodynamics of the Stars. Handbuch der Astrophysik, Vol. 3, Part 1. 

This book contains an elegant deduction of Schwarzschild’s (1906) plane-parallel (“linear”, “tubular”) solution from Schwarzschild’s (1914) general (differential and integral) equations.

Schwarzschild (1906), (1914) and Milne (1930) was translated to English and published in Selected papers on the transfer of radiation, Donald H. Menzel, 1966. Dover Publ.

Goody, R. (1964) Atmospheric radiation: Theoretical basis. Oxford University Press.

Contains the deduction of the constant net radiation at the surface as equals half of the incoming solar (and outgoing longwave) flux Fs (Equation 2.115), noting that the jump always equals Fs/2.

Manabe, S. and Möller, F. (1961) On the radiative equilibrium and the heat balance of the atmosphere. Monthly Weather Review, 89:12

Refer to Emden (1913), but not to Schwarzschild.

Manabe, S. and Strickler, R. (1964) Thermal equilibrium of the atmosphere with a convective adjustment. Journal of the atmospheric Sciences, 21.

Refer to Emden, but not to Schwarzschild. Compute correctly the magnitude of the temperature discontinuity, but do not mention the constraint on it.  

Manabe, S. and Wetherald, R. (1967) Thermal equilibrium of the atmosphere with a given distribution of relative humidity. Journal of the atmospheric Sciences, 24.3.

Refer neither to Emden nor to Schwarzschild.  Compute correctly the magnitude of the temperature discontinuity, but do not mention the constraint on it.

Houghton, J. (1977) The physics of atmospheres. Cambridge University Press. 2nd and 3rd editions 1986, 2002.

Equation 2.13 reproduces the net radiation at the surface as equals half of the emerging flux. Notes that the discontinuity in temperature in radiative equilibrium is destroyed by the process of convection, leading to radiative-convective equilibrium.

Chamberlain, J. (1979, 2nd ed. 1987) Theory of planetary atmospheres. Academic Press.

Fig. 1.4 explicitly shows the discontinuity as F/2.

Hartmann, D. (1994, 2nd ed. 2016) Global physical climatology.

For its three-layer radiative equilibrium model, Eq. 3.51 gives the surface temperature and Eq. 3.54 the temperature of the air adjacent to the surface. Their difference is not displayed explicitly, but directly follows as σTS4 – σTSA4 = σTe4 /2. With its data, the difference is 0.31 Wm-2.

Salby, M. (1996) Fundamentals of atmospheric physics. Academic Press. 2nd ed. 2012, Cambridge Univ. Press.

Eq. 8.67 gives the net radiation at the surface equals F0/2.

Visconti, G. (2001) Fundamentals of physics and chemistry of the atmosphere. Spinger Verlag.

Eq. 3.49 gives the formula for the net radiation at the surface as half of the effective emission.

Vardavas, I. and Taylor, F. (2007) Radiation and Climate. Oxford Univ. Press.

“An atmosphere in radiative equilibrium (see Fig. 2.11) produces essentially a discontinuity (of about 20 K) between the Earth’s surface temperature and the near–surface atmospheric temperature”, equals f/2.

Zdunkowski, W., Trautmann, T. and Bott, A (2008) Radiation in the Atmosphere. Cambridge Univ. Press.

Fig. 6.7 shows the discontinuity.

Pierrehumbert, R. (2008) Principles of Planetary Climate. Cambridge Univ. Press, Eq. 4.45

Andrews, D. (2010) An Introduction to Atmospheric Physics. Cambridge Univ. Press. Pp.85-86, Fig. 3.21.

Ambaum, M. (2021): Thermal physics of the atmosphere. Royal Meteorological Society. 

Eq. 10.56. Temperature discontinuity, removed by turbulent heat exchange.

 

The theory is straightforward; is it true on observations?

CERES EBAF Edition 2.8 global means 2000-2016:

Net radiation at the surface = TOA LW up/2 – 0.59 Wm-2. 

CERES EBAF Edition 4.1 global means 2000-2024:

Net radiation at the surface = TOA LW up/2 – 2.33 Wm-2. 

All-sky:

Net radiation at the surface = (TOA LW up – LWCRE)/2 + 2.71 Wm-2. 

Mean bias of the two net equations: 0.19 Wm-2.

All-sky version of the equation is valid on 30 years of GEWEX observation dataset with a difference of 0.1 Wm-2.

 Stephens, G. et al. (2023) The first 30 years of GEWEX. BAMS

References where the equation is not there:

All the follow-ups of the Manabe-approach

The Charney report (1979)

Theory of Climate (1980) (ed. B. Saltzman). Academic Press

IPCC reports (1990-2021).

For example, an ideal position to include it would have been in WGI AR5 (2013), Chapter 2.3 (Changes in Radiation Budgets), where said: “On average, radiative processes warm the surface and cool the atmosphere, which is balanced by the hydrological cycle and sensible heating.” Here to declare the constraint on the hydrological cycle and sensible heating as OLR/2 in clear-sky is necessary.

NASA CERES Science page, Global Mean Energy Budget

Page created by: Miklos Zagoni

Contact: miklos.zagoni@earthenergyflows.com

IUS ERRORIS RESERVATUR