Development in Earth Science Volume 2, 2014

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The Greenhouse Effect and the Infrared
Radiative Structure of the Earth's Atmosphere
Ferenc Mark Miskolczi
Geodetic and Geophysical Institute, Hungarian Academy of Sciences, Csatkai Endre u. 6-8, 9400 Sopron, Hungary
fmiskolczi@cox.net
Abstract

FE  FA  FR and FA  OLR A relationships must hold,

This paper presents observed atmospheric thermal and
humidity structures and global scale simulations of the
infrared absorption properties of the Earth's atmosphere.
These data show that the global average clear sky greenhouse effect has remained unchanged with time. A theoretically predicted infrared optical thickness is fully
consistent with, and supports the observed value. It also
facilitates the theoretical determination of the planetary
radiative equilibrium cloud cover, cloud altitude and Bond
albedo. In steady state, the planetary surface (as seen from
space) shows no greenhouse effect: the all-sky surface upward radiation is equal to the available solar radiation. The
all-sky climatological greenhouse effect (the difference of the
all-sky surface upward flux and absorbed solar flux) at this
surface is equal to the reflected solar radiation. The planetary radiative balance is maintained by the equilibrium
cloud cover which is equal to the theoretical equilibrium
clear sky transfer function. The Wien temperature of the allsky emission spectrum is locked closely to the thermodynamic triple point of the water assuring the maximum
radiation entropy. The stability and natural fluctuations of
the global average surface temperature of the heterogeneous
system are ultimately determined by the phase changes of
water. Many authors have proposed a greenhouse effect due
to anthropogenic carbon dioxide emissions. The present
analysis shows that such an effect is impossible.

where FA is the long term global average absorbed SW

Keywords
Greenhouse Effect; Radiative Transfer; Global Warming

Introduction
In steady state planetary radiative balance the relationship that links the short wave (SW) solar radiation
to the long wave (LW) terrestrial or infrared (IR)
radiation may be expressed as : OLR A  (1   B ) FE .
Here OLR A is the all-sky outgoing LW radiation,
 B  FR / FE is the Bond albedo, FR is the all-sky
reflected SW radiation, FE  F0 / 4 is the available SW
radiation over a unit area at the top of the atmosphere
(TOA), and F0 is the solar constant at the Earth's orbit.
Isolated planets without any internal heat source must
obey the energy conservation principle, therefore the

radiation in the system. The balance relationship does
not suggest anything about how, when, and why the
observed thermal energy of the planet is attained
during the evolution of the planet. The only meaning
of the balance equation is the equality between the
thermal energy lost to space and the gained radiative
energy by SW absorption when a steady state has been
reached.
In a planetary atmosphere with condensing greenhouse gases (GHGs), the active surface that is relevant
to the radiative balance equation is the combined clear
and cloudy surfaces as seen from space. The ratio of
the overcast areas to the total surface area of the planet
is called geometric cloud fraction,  . Because of the
extreme variability of the planetary cloud cover, the
accurate estimation of  from surface or satellite observations is one of the most challenging problems of
climate science. The characteristic global average altitude of the cloud top, h C , is also not known with very
high accuracy. Missing from climate science literature
are the quantitative theoretical constraints on the
value of the  and h C parameters. These issues will be
discussed later in detail.
Cloud layers at any altitude present material discontinuity in the atmospheric vertical structure which
disrupts the propagation of the LW radiation. In
principle, the global average LW upward radiation
from the ground surface, SU , may be estimated
reasonably well from ground surface temperature
records. Over cloudy areas, however, the upward IR
flux density from the cloud top, SUC , cannot be easily
measured. The SU from overcast areas does not contribute to the total OLR A : OLR A  (1 ) OLR   OLR C ,
where OLR is the clear sky, and OLR C is the cloudy sky
outgoing LW radiation. Estimations of OLR C from a
unit cloudy area must rely on the observed h C ,

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Development in Earth Science Volume 2, 2014

therefore they are inherently inaccurate. The radiative
equilibrium constraint for the clear and cloudy areas
are expressed as: SU  OLR / f  OLR (1   A  TA ) / 2 ,

component in the cloudy atmosphere is the total
upward radiation (transmitted from the surface plus
emitted by the atmosphere) at the altitude of the cloud

 OLR C (1   AC  TAC ) / 2 , where

bottom: OLR C u  STC u  EUC u . The absorbed surface

and SUC  OLR C / f

C

 A , TA , and f are the flux optical thickness, flux
transmittance, and transfer function respectively, and
the superscripts indicate the cloudy condition, see
Miskolczi (2004, 2007, and 2010). Further on we shall
frequently make reference to these publications as
M04, M07, and M10. The breakthrough in the
quantitative greenhouse science happened in 2007,
when the correct mathematical relationship among SU ,

upward flux at the cloud bottom is AA C u  SU  STC u .
Greenhouse Effect
The planetary greenhouse effect (GE) may be defined
or quantified in different ways. In astrophysics the allsky GE is defined via the total available solar radiation
interacting with the system: G A  SUA  FE , where

OLR , and  A for semitransparent atmospheres was

SUA  (1   ) SU   SUC is the all-sky global average

first published in M07. The theoretical derivation of
the SU  OLR / f analytical function was the missing

surface upward flux (from the active surface).

link which, through the transfer function and flux
optical thickness, connects the surface temperature to
the GHG content of the atmosphere.

weighted sum of the clear sky and cloudy sky

Here one has to be careful with the computation of SU
from the related ground surface thermodynamic
temperature, tG . For non-black surfaces, the upward
radiation is defined by the skin temperature tS :

Similarly to SUA , G A may be obtained from the
greenhouse effects: G A  (1   ) G   G C , where G 

SU  OLR , and G C  SUC  OLR C .
In a semi-transparent clear atmosphere, OLR is the
sum of the transmitted flux density from the surface,
ST and the atmospheric upward emittance, EU :

 SG , where  G is the surface flux

OLR  ST EU . In a clear, absorbing GHG atmosphere,
SU  ST  AA and from the definition of G , follows the

emissivity,   5.67 108 Wm-2K-4 is the StefanBoltzmann constant. SG  B(tG )  BG is the total flux

SU  OLR  AA  EU greenhouse identity. Here AA is
the clear sky absorbed SU . Likewise, the greenhouse

density radiated into the hemisphere from an ideal
blackbody radiator at tG temperature. In planetary

identity for the fluxes above the cloud top is
SUC  OLR C  AAC  EUC , where AAC is the absorbed SUC

SU  

tS4

  G

tG4

radiative budget studies  G  1 , and SU  SG are usually

above the cloud layer and EUC is the upward emission

assumed. For reference, in Trenberth, Fasullo, and
Kiehl (2009) (TFK09) the broadband emissivity of the
water is  G  0.9907 and the ISCCP-FD tG is 288.70 K.

of the air from above the cloud top. For all-sky fluxes,
the SUA  OLR A  AAA  EUA relationship must be

The clear and cloudy sky LW downward atmospheric

cloudy sky, STC , transmitted fluxes are: STA  SUA  AAA ,

emittance to the ground surface and cloud top are ED

and STC  SUC  AAC , respectively. AAA  (1   ) AA   AAC
is the all-sky absorbed LW radiation (above the active
surface). In accurate planetary radiative transfer (RT)
computations, the greenhouse identities must be
observed. The normalized all-sky, clear sky and
cloudy sky greenhouse factors (GFs) are the
g A  G A / SUA , g  G / SU , and the g C  G C / SUC ratios

and

EDC

respectively. The all-sky LW downward flux

to the ground surface is the sum of the clear and
cloudy components: EDA  (1   ) ED   OLR Cd , where

OLR C d  STC d  EDC d is the cloudy sky contribution.
Here STC d is the downward transmitted flux from the
cloud bottom, and EDC d is the downward atmospheric
emission from below the cloud layer.

SDC

is the

downward radiation emitted by the cloud bottom, and
by definition, AAC d  SDC  STC d is the absorbed part of

SDC . Assuming a thin opaque cloud layer the upward
and downward radiation (emitted by the cloud deck)
are equal, SDC  SUC . The last important flux density

32

satisfied as well. By definition, the all-sky, STA , and the

respectively. Here g A is not a simple weighted average: g A  g / (1 SUC / (1  ) / SU )  g C / (1(1 ) SU / /SUC ) .
It is anticipated that the thermal structure of the
atmosphere is always affected by both the local, and
the ever-present global average cloud cover. In terms
of the quasi all-sky protocol, clear sky computations of
the flux density components are conducted ignoring

Development in Earth Science Volume 2, 2014

the possible (random) presence of the cloud cover. The
computed global average fluxes are assumed to
implicitly represent the global average cloud condition. In fact, in climate science the classic definition
of the greenhouse warming (as the difference of the
all-sky global average surface temperature and the
planetary emission temperature) is a version of the
quasi all-sky protocol.
In the case of planetary radiative equilibrium the
global average net energy flux of non-radiative origin
(conduction, convection, advection, turbulent mixing,
etc.) between the solid and liquid surfaces and the
atmosphere must be zero. Of course, the net latent
heat release at the boundary layer must be treated as
of radiative origin. In global radiative equilibrium

SUA  FE , OLR A  FA , G A  FR , g A   B ,   f ( AT ) ,

SU  OLR / f ( AT ) , OLR  AAC , and OLR C  EDC , where

 AT  1.876 is the planetary equilibrium flux optical
thickness. Note, that the last four relationships are
derived theoretically in M07. It is assumed, that the
equilibrium atmospheric structure is such, that the
cloud cover alone is able to maintain the planetary
radiative balance. At the TOA one may write the
theoretical equilibrium cloud fraction and the Bond
albedo in the simple forms of   (G B FE ) / (GG C ) ,
and  B  (G   (G  G )) / FE .
C

In climate science, the all-sky and clear sky greenhouse parameters are defined through the OLR ,

OLR A , and SG fluxes: GmA  SG  OLR A , Gm  SG  OLR ,

g mA  GmA / SG , and gm  Gm / SG . Gm and g m are the
clear sky GE and GF, Ramanathan and Inamdar (2006)
(RI06). Sometimes g m is called as 'normalized trapping'
of SG . We have seen already, that without a realistic

 G the tG temperature alone is not sufficient to convert
SG to accurate SU . Further on, one should notice, that

GmA , Gm , gmA , and g m are mixed physical quantities and
they cannot be associated with either clear, or cloud
covered surfaces. To handle the cloud problem, the socalled LW cloud forcing as the difference of the clear
and all-sky TOA terrestrial radiation is also introduced:
CL  OLR  OLR A . The frequently used total greenhouse
effect terminology in climate science means that
GmA  Gm  CL  SG  OLR A . Although both SG and

OLR A may easily be observed, GmA cannot be related
directly to the GHG composition of the atmosphere,
and it has no clear physical meaning. The cloud cover

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has nothing to do with the absorption of the GHGs.
Water droplets just like other solid or liquid surfaces
radiate continuous IR spectra. The greenhouse effect
from the GHGs above the cloud layers should also be
taken into account.
The GE based anthropogenic global warming (AGW)
hypothesis rests on the assumption that increasing
atmospheric CO2 concentration of human origin will
result in increasing global average ground surface
temperature. The motivations for writing this paper
are the accumulating evidences that GE in the Earth's
atmosphere is not a free parameter. AGW estimates
based on the classic greenhouse effect explanations of
Fourier, Arrhenius, Tyndall, or the Intergovernmental
Panel on Climate Change (IPCC) are misleading. The
well known, and widely used, semi-infinite opaque
formulas with their predicted surface temperature
discontinuity cannot be used for semi-transparent
atmospheres. AGW predictions, based on modelling
calculations for CO2 doubling, are also not consistent
with the observed global average surface temperature
records of the last decades. Furthermore, observed
local or regional warmings that are usually attributed
to greenhouse warming (like Arctic warming) may be
accounted for by quite natural causes, see Arrak (2010,
2011).
In this paper, the published atmospheric greenhouse
effect and global warming related articles that
appeared in the climate science literature, are not
reviewed. There are excellent articles summarizing the
level of the general understanding (or misunderstanding) of the phenomenon, see Herzberg (2009),
Kimoto (2009), Gerlich and Tscheuschner (2009), Van
Andel (2010), Hansen et al. (1981), Ramanathan (1981),
Raval and Ramanathan (1989) , Lindzen, (2007), Lacis
et al. (2010), and Pierrehumbert (2011). The usually
quoted quantities are the GmA   tG4   t A4  150 Wm-2
all-sky GE and the  tmA  tG  t A  33 K greenhouse
warming. Here tG  288 K is the all-sky global average
ground surface temperature,  tG4  390 Wm-2 is the
surface upward radiation, tE  ( FE /  ) 1/4  279 K is the
planetary effective (or equivalent blackbody) temperature t A  (OLR A /  )1/4  255 K is the planetary emission
temperature, FE  342 Wm-2, and OLR A  239 Wm-2.
Note that the astrophysical GE is much smaller than

GmA . Among the authors, there seems to be an
agreement that the absorption and re-emission of the
surface upward infrared radiation by GHGs are the

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principal causes for greenhouse warming. And, due to
well established energy balance principles, increased
atmospheric CO2 concentration will be inevitably
followed by an increased greenhouse effect. However,
the fact is that the greenhouse effect is a differential
quantity, therefore, such statements are not very well
established and demonstrated. We have seen that Gm is
a mixed quantity and cannot be associated with the
net absorption of the SU in the system.

Development in Earth Science Volume 2, 2014

Climate modelers are using diverse - and not very
transparent - H2O feedback processes to match their
predicted OLR with the reality. Here OLR is the
difference between the OLR of the unperturbed and
perturbed cases. In this article the High Resolution
Atmospheric Radiative Transfer Code (HARTCODE)
line-by-line (LBL) RT software are used for all flux
computations. Typically, the spectral resolution was
set to 1 cm-1, Miskolczi (1989) .

The most popular global energy budget schemes were
published by Kiehl and Trenberth (1997) (KT97), and
TFK09. In KT97 and TFK09 the global average
terrestrial radiation field was modeled by using a
version of the US Standard Atmosphere 1976 (USST76)
in which in order to match with Earth Radiation
Budget Experiment, ERBE (2004) observations the H2O
column amount was reduced from 1.42 to 1.26
precipitable cm (prcm). In KT97 and TFK09 the all-sky

GmA and clear sky Gm were reported as 155 and 157
Wm-2, subsequently. Newer studies show substantially different greenhouse effects, see Stephens et al.
(2012) (S12), Wild et al. (2013) (W13), and Costa and
Shine (2012) (CS12). In Lacis et al. (2010) GmA is less by
5 Wm-2 than the one in RI06. The Gm in M10 shows
about 10 Wm-2 underestimate, compared to the one in
Lacis et al. (2010). Such differences in the radiative
fluxes may be translated into about 1-2 K uncertainties
in tE and tG temperatures. In Fig. 1, we show how CO2

FIG. 2 SPECTRAL FLUX DENSITY COMPONENTS IN COLD AND
DRY, (LEFT PLOT), AND WARM AND HUMID, (RIGHT PLOT),
SITUATIONS. ED : DOWNWARD ATMOSPHERIC EMITTANCE;
G : CLEAR SKY GREENHOUSE EFFECT;  E : RADIATIVE EQUI-

LIBRIUM FLUX OPTICAL THICKNESS. FLUXES ARE IN WM-2.

perturbations affect the OLR and how the real world
responds to the changeing atmospheric CO2 amount.

FIG. 3 SPECTRAL FLUX DENSITY COMPONENTS IN THE
GLOBAL AVERAGE NOAA R1, (LEFT PLOT), AND USST76 ,
(RIGHT PLOT), ATMOSPHERES. FLUXES ARE IN WM-2 .

FIG. 1 HARTCODE GHG PERTURBATION STUDY SHOWS THAT
AT THE TOA THE NO-FEEDBACK RESPONSE OF INCREASED
ATMOSPHERIC CO2 IS NEGATIVE. THE OBSERVED 23.6 %
INCREASE IN THE CO2 COLUMN AMOUNT CAUSES -0.75 WM-2
RADIATIVE IMBALANCE (RED DOT). IN THE SAME TIME
PERIOD, BASED ON THE NOAA R1 ARCHIVE THE REAL
CHANGE IS OLR  3.02 WM-2 (BLUE DOT).

34

In Figs. 2 and 3 the theoretical difficulties of the
interpretation of G as the measure of the absorption
properties of the atmosphere have been demon-strated.
In Fig. 2 local clear sky greenhouse effects are
computed and compared for cold, and also for warm
real atmospheric structures. The  A IR flux optical
thicknesses are very similar in the two cases. The H2O

Development in Earth Science Volume 2, 2014

column amounts and the radiative equilibrium optical
thicknesses,  E , are largely different and are consistent
with the H2O amounts. Because of the chaotic nature
of the humidity field and cloud cover, one cannot
quantitatively relate the local G or g to the GHG
content of the atmosphere.
In the second example (see Fig. 3) computations are
performed using the 61 year global average
atmosphere from the NOAA NCEP/NCAR (2008)
reanalysis data time series (NOAA R1), and the
version of USST76, used in KT97 and TFK09. Note that
the H2O column amounts in the two profiles are
dramatically different. In the above example, the
global average G and g are not sensitive to the roughly
doubled water vapor amount in the atmosphere.
These comparisons clearly show that the greenhouse
effect characterized with Gm or G is not consistent with
statements that link the increased GHG content of the
atmosphere to increased IR absorption.
Considering the above examples, and the permanent
failure of the most sophisticated general circulation
models (GCMs) in predicting the magnitude of global
warming, one should admit the serious theoretical
deficiencies in using the greenhouse effect as a sole
measure of infrared atmospheric absorption. The
governing mechanisms of the IR absorption properties of the global average atmosphere are never
studied in sufficient detail and the real nature of the
greenhouse effect is not known. Compared to the
observed ~0.012 K/year positive trend in the surface
temperature over the last 61 years (see M10) and the
recent skills of the GCMs in predicting the changes in
the GE for a hypothetical CO2 doubling, the quantitative proof of the CO2 greenhouse effect based AGW
is not imminent.
In any serious greenhouse study, the knowledge of the
functional dependence of the global average IR flux
optical thickness on the GHG concentrations, and the
surface temperature, is absolutely necessary. The flux
optical thickness,  A , flux absorption, A , and flux
transmittance, TA , are defined by the ST  SU exp( A )
and ST  SU (1  A)  SU TA relationships. For semitransparent atmospheres, except in M04, M07 and M10,
there are no published numerical data available on the
theoretical surface temperature - flux optical thickness
relationship. The obvious reason why the scientific
community does not present such results is twofold.
The first is the lack of a suitable greenhouse theory
which is based solely on the known fundamental laws

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of nature. Apart from the fact, that the use of GCMs
for studying large scale climate change is conceptually
wrong (fundamentally stochastic processes cannot be
diagnosed by a deterministic model), the GCMs with
their numerous tuning parameters are not representing the principles of physics and the demonstrated
response of the greenhouse effect. Common greenhouse effect explanations are not able to account for
the magnitude and the tendency of the phenomenon.
It has been known for a long time that climate change
is controlled by the net radiative fluxes at the TOA and
at the ground surface. The global average state of the
atmosphere (or global average climate) is governed by
the laws that control flows of radiative fluxes at the
boundaries.
The second reason is rather technical, and related to
the accurate computation of the flux optical thickness.
According to RI06 the three dimensional characterization of radiative heating rates from equator to
pole using the LBL approach is impractical. This view
suggests sacrificing accuracy, by using band models in
global scale radiative transfer computations, where it
is most needed. This simplified view is probably the
reason why, in recent textbooks, extensive parts are
devoted to popularizing ancient band model techniques, see for example in Pierrehumbert (2010).
Unfortunately the fact is that there are no publicly
available LBL codes for accurate computations of the
IR flux optical thickness. From a correct LBL spectral
radiance code there is a very long way to a correct
spectral spherical refractive flux density code.
The IR Optical Thickness of the Atmosphere
In astrophysics - for the different kind of radiative
transfer problems - there are different kinds of
definitions for the mean optical thickness (or mean
opacity), see Mihalas and Weibel-Mihalas (1999). They
are the Rosseland, Planck, and Chandrasekhar means,
and they are, in fact, different kinds of weighted
average absorption coefficients. The relevant physical
quantity necessary for the computation of the real
atmospheric IR absorption is the Planck-weighted
greenhouse-gas optical thickness,  A . The numerical
computation of this quantity for a layered spherical
refractive atmosphere may be found in M10. By
definition,  A is computed from the spectral hemispheric transmittance and therefore represents the real
spectral feature of the infrared absorption coefficient.
It should be emphasized that  A is not a weighted
absorption coefficient in the sense of the usual Planck

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Development in Earth Science Volume 2, 2014

mean opacity in Mihalas and Weibel-Mihalas (1999).
 A is a newly defined physical quantity and one
cannot find any reference in the literature to its
computational techniques. The existence of the large
and organized absorption line catalogues and the
development of high speed computers and LBL
computational techniques are the reasons for the
above definition of  A , see HITRAN2K (2002),
Rodriguez et al. (1999). Only a full blown spherical
refractive LBL radiative transfer code is able to
compute the accurate atmospheric IR flux optical
thickness. In short,  A may be expressed as:
K
 1 M
k
k 
  B ( j , tG )  w TA (  j ,  )  ,
4
j

1
k

1
  tG


 A   ln 

(1)

where M  3490 is the total number of spectral
intervals, K  9 is the total number of streams,
B ( j , tG ) is the mean spectral Planck function, and

w k is the hemispheric integration weight associated
with the k t h direction (stream), TA ( j ,  k ) is the direc-

FIG. 4 DOWN-LOOKING CASE. HARTCODE SPECTRAL HEMISPHERIC TRANSMITTANCES IN THE 1-3490 CM-1 SPECTRAL
INTERVAL FOR THE 668 CM-1 INTERVAL THE DIRECTIONAL
TRANSMITTANCES ARE ALSO PLOTTED (WITH GREEN LINES).
THE BLACK DOTS REPRESENT THE ISOTROPIC ANGLE AND
INDICATE CONSIDERABLE ERROR IN THE WIDELY USED
ISOTROPIC APPROXIMATION.

tional mean spectral transmittance over a suitable
short wave number interval:
 L N
u i ,l 
TA ( j ,  k )   j 1  exp     c i ,l  ki ,l  l , k  d , (2)
 

 l 1 i 1
j

where  l ,k  cos( l ,k ) / dz l and  l ,k is the local zenith
angle of a path segment, c i ,l and ki ,l are the contributions to the total monochromatic absorption
coefficient from the continuum type absorptions and
all absorption lines relevant to the i th absorber and l th
layer respectively. The vertical geometric layer
thickness is dz l . N  11 is the total number of major
absorbing molecular species and L  150 is the total
number of the homogeneous atmospheric layers
(shells). In Eqn. (2) the wave number integration is
performed numerically by fifth order Gaussian
quadrature over a wave number mesh structure of
variable length. At least  j  1 cm-1 spectral resolution
is required for the accurate Planck weighting. From
Eqn. (1) follows the usual form of the transmitted and
absorbed part of the surface upward radiation. Eqs.
(1,2) with the required spherical refractive ray-tracking
algorithms are implemented into HARTCODE and
facilitate the accurate partition of the OLR to its ST and

EU components. The oversimplified and, in fact, often
mathematically incorrect computation of Gm (for example in CS12, RI06, or in the NATURE article of
Raval and Ramanathan (1989)) should be avoided.

36

FIG. 5 HARTCODE SPECTRAL HEMISPHERIC TRANSMITTANCES IN THE 1-3490 CM-1 SPECTRAL INTERVAL. THESE TRANSMITTANCES ARE REQUIRED FOR THE COMPUTATION OF THE
DOWNWARD FLUXES FROM THE TOA OR FROM THE CLOUD
BOTTOM.

Unfortunately, theoretically, no instrument can be devised to measure the monochromatic or spectral ST
and EU quantities separately. Since the above radiative
components cannot be measured by any airborne or
satellite spectrometer this is an essential improvement
in the numerical computations of the real IR atmospheric absorption.
In Fig. 4 and Fig. 5 hemispheric transmittances
(obtained from Eq. (2) by integration over the
respective hemispheres) are presented for the global

Development in Earth Science Volume 2, 2014

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average Thermodynamic Initial Guess Retrieval (GAT)
profile and for down-looking and up-looking
geometries. According to the Helmholtz reciprocity
principle the spectral mean downward and upward
hemispheric transmittances must be equal. In M10,
page 234, the atmospheric transfer and greenhouse
functions were introduced by the f ( A )  2 / (1  A TA ) ,
and g ( A )  1  f ( A ) definitions. For an atmospheric
air column in radiative equilibrium it was also shown
that f  OLR / SU , and g  (SU  OLR) / SU . Using the
relationships above, the normalized greenhouse factor
may be expressed with TA , A , or  A . The complex,
nonlinear

dependence

of g

on

the

absorption

properties of the atmosphere and the boundary layer
fluxes is apparent. One must remember that the so
called broadband window radiation is not an adequate
quantity to represent the transmitted surface radiation.
To make use of the satellite measured global average
broadband window radiation in global radiative
budget estimates the data should be corrected (or
calibrated) with global average atmospheric absorption data of the highest accuracy. Previously, in
Fig. 3 accurate clear sky transmitted flux densities are
presented for the GAT and the USST76 atmospheres.
The 30 Wm-2 difference in ST is large enough to raise
the question of the quality of the KT97 and TFK09
global energy budgets. Although the USST76
atmosphere could be a good representation of an
average mid-latitudinal atmospheric structure, the use
of it in global energy budget assessments is a serious
mistake.
It must be recognized that no consensus in global
warming issues can exist without a declared and
accepted standard global average atmosphere derived
from a well documented global radiosonde archive.
The total IR absorption of such an atmosphere must be
computed for the most realistic chemical and GHG
composition of the atmosphere and with the highest
accuracy. All GHG studies and radiative budget
estimates should be referenced to the absorption and
optical properties of this global average standard
atmosphere. In looking for flux density - optical
thickness relationships, which could be used for
surface skin temperature estimates, or in the quantitative computations of the GE sensitivities in GHG
perturbation studies, such an atmosphere would be
extremely beneficial. As an example, it has little merit
to reference to RT computations where the atmospheric thermal and humidity structure (or the related

input data base) are not traceable to the original
sources. In the weakly documented low quality paper
of CS12, no one knows how much water vapor or CO2
is in the air, yet they suggest an accurate global
average ST for other people to use in their energy
budget studies , see S12.
Although, in some cases, empirical estimates of radiative budget components from other authors are
referenced, the critical evaluation of the different
planetary radiative budget schemes is not the purpose
of the present article. In general, in science, a debate
over an issue is initiated when the related subject is
sufficiently well known and both theoretical and
empirical supports are available for the discussion. For
example, no one will seriously comment upon the
fictitious surface energy imbalance of 0.6  17 Wm-2 in
S12, or the 0.6  0.4 Wm-2 in W13.
In the next sections we present numerical results of
observed radiative flux density relationships for the
planet Earth, identify and develop the theoretical
relationships consistent with the observations and give
a new view of the planetary greenhouse effect.
Input Data Sets
Realistic vertical global average thermal and humidity
structures may be obtained from readily available
climatological radiosonde archives, Chedin and Scott
(1983). In this study the GAT global average structure
is constructed from the Thermodynamic Initial Guess
Retrieval (TIGR2) archive containing 1761 weather
balloon observations. An updated version of the
database (known as the TIGR2000 archive) containing
2311 soundings is also available: TIGR Thermodynamic Initial Guess Retrieval (2000). The locations,
meridional, and annual distributions of the two
archives are presented in Fig. 6. Both archives contain
prohibitively large number of soundings for LBL
computations.
One should be aware of some inconsistent and
undocumented modifications of the vertical humidity
structures in the TIGR2000 soundings. In hundreds of
TIGR2000 observations the upper tropospheric humidity is significantly increased which may introduce
biases in the computed global average vertical
radiative structure. After some regional and seasonal
grouping of the TIGR2 soundings a subset of 228
profiles is selected, see Fig. 7. In the subset the
statistical characteristics of the original data set are
preserved. In Fig. 8, two extreme atmospheric structures from the selected sub-set are presented.

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In this article we use the GAT atmosphere as the
repre-sentative temperature and humidity structures
of the global average climate. For studying possible
long term changes in the global average optical
thickness (due to changes in GHG content of the
atmosphere) the TIGR2 archive is not suitable. The
publicly available longest time series of annual mean
vertical temperature and humidity structures may be
obtained from the NOAA R1 time series data archive.
This archive, known as the NCEP/NCAR R1 data set,
covers the 1948-2008 time period and is regularly

Development in Earth Science Volume 2, 2014

updated. NOAA R1 has been used by the NCEP
Climate Prediction Center to produce global
atmospheric monitoring and assessment products,
Trenberth (2009). A quick look at the data immediately
shows that the range of variations in the annual mean
over 61 years is very small: 58.87 atm-cmSTP in CO2,
0.0169 prcm in H2O, and 0.687 K in surface temperature. The related year-to-year relative changes are
also small, 0.35 %/year in CO2, -0.0106 %/year in H2O,
and 0.0039 %/year in surface temperature. In this
study monthly or seasonal variations are ignored.

FIG. 6 THE TIGR CLIMATOLOGICAL DATASETS. DETAILED COMPARISONS SHOW
THAT THE GLOBAL AVERAGE TIGR2 SURFACE AIR TEMPERATURE IS 0.28 K COLDER
AND THE VERTICAL AIR COLUMN CONTAINS 0.1 PRCM (ABOUT 3 %) LESS H2O.
SINCE IN THE TIGR2000 VERSION THE VERTICAL H2O STRUCTURE WAS ARTIFICIALLY MODIFIED (UPPER TROPOSPHERIC HUMIDITY WAS INCREASED) WE
DECIDED TO USE THE ORIGINAL TIGR2 ARCHIVE.

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Development in Earth Science Volume 2, 2014

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Obviously, there are very high requirements for the
sensitivity and numerical accuracy of the computed
fluxes and flux optical thicknesses. To quantify the
possible trend in the average absorption properties of
the atmosphere, simulations are performed for six
subsets of different times and time intervals.
Observed Empirical Facts
In 2002, at the National Atmospheric and Space
Administration (NASA) Langley Research Center, the
first set of global scale high accuracy LBL flux optical
thickness and flux density computations for the TIGR2
data set were completed, and the new fundamental
clear sky semi-transparent radiative equilibrium equation, SU ( A )  OLR( A ) / f ( A ) , was awaiting large scale
empirical verification. A general view of the simulated
flux density components are presented as the function
of the 11 seasonal and geographical classes in Fig. 9. A
quick look at the upper plot immediately confirms the
obviously expected relationships among the fluxes:
ST  EU  OLR  ED  AA  SU . The lower plot suggests a fairly strong linear relationship between the
downward ED and the absorbed AA fluxes. After the
routine plots of  A , TA , SU , ST , ED , EU , and OLR
quantities, four rather unusual relationships among
the flux density components and optical thicknesses
emerge. In Figs. 10-13 the TIGR2 simulation results are
plotted for the individual soundings. The tentative
naming of the discovered empirical relationships (they
are called 'rules') reflects the fundamental physical
laws with which they are associated. For completeness, we should also mention the extropy rule, which
is not discussed here, see Miskolczi (2011).

FIG. 7 LATITUDINAL AND SEASONAL GROUPING OF THE
TIGR2 SOUNDINGS. IN THE SELECTED SUBSET 228 SOUNDINGS
WERE DISTRIBUTED AMONG 11 GROUPS HAVING ABOUT 20
SOUNDINGS IN EACH GROUP. LATITUDINAL AND SEASONAL
CLASSES WERE ESTAB- LISHED CONSIDERING THE SOLAR
CLIMATIC ZONES.

FIG. 8 RARE ATMOSPHERIC SITUATIONS. EXTREME DRY AND
COLD AND WARM AND HUMID ATMOSPHERIC STRUCTURES
IN THE TIGR2 DATA .

FIG. 9 SEASONAL-GEOGRAPHICAL DISTRIBUTIONS OF THE
TIGR2 FLUX DENSITIES. CLASSES ARE: 1 - ARCTIC SUMMER, 2 ARCTIC WINTER, 3 - NORTH MID-LATITUDE SUMMER, 4 NORTH MID-LATITUDE FALL/SPRING, 5 - NORTH MIDLATITUDE WINTER, 6 - NORTH/SOUTH TROPICAL, 7 - SOUTH
MID-LATITUDE SUMMER, 8 - SOUTH MID-LATITUDE
FALL/SPRING, 9 - SOUTH MID-LATITUDE WINTER, 10 ANTARCTIC SUMMER, 11 - ANTARCTIC WINTER. THE LOWER
PLOT SHOWS THAT IN ALL CLASSES THE ATMOSPHERIC
ABSORPTION CAN BE VERY WELL APPROXIMATED BY THE
DOWNWARD ATMOSPHERIC EMISSION.

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FIG. 10 THE ATMOSPHERIC KIRCHHOFF RULE SUGGESTS A
VERY STRONG LINEAR RELATIONSHIP BETWEEN THE
EMITTED CLEAR SKY DOWNWARD AND THE ABSORBED
SURFACE UPWARD IR FLUX DENSITIES. COLD AND DRY
POLAR AREAS FIT BETTER TO THE RULE THAN THE WARM
AND HUMID EQUATORIAL AREAS. THE GLOBAL MEAN
CLEAR SKY ED / AA  1 RATIO (WE CALL IT SPHERICAL
EMISSIVITY) IS ABOUT 3 %.

FIG. 11 THE RADIATIVE EQUILIBRIUM RULE IS THE NAME OF
THE THEORETICALLY DERIVED OLR / SU  f EQUATION. FOR
THE TIGR2 ARCHIVE THE GLOBAL AVERAGE TRANSFER
FUNCTION IS SLIGHTLY LARGER THAN THE THEORETICALLY
EXPECTED f  0.6618 . THE SCATTER OF THE POINTS SHOWS
THAT THE EXACT LOCAL RADIATIVE EQUILIBRIUM IS A VERY
RARE SITUATION.

At this point we should emphasize that the presented
relationships are not derived from some well known
physical laws of nature, but are obtained from observations and computations using first principles. It
should also be noted that the newly discovered
empirical relationships are not the results of some
lucky coincidental profile selections from the TIGR2
archive. In the last several years the computations
using the TIGR2000 archive, the updated NOAA R1
archive, some satellite training and calibration data
sets, hundreds of high resolution and high quality

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Development in Earth Science Volume 2, 2014

FIG. 12 THE SU  3OLR / 2 RELATIONSHIP SHOWS GOOD AGREEMENT BETWEEN THE GLOBAL MEAN (BLACK CIRCLE) AND
THE THEORETICAL EXPECTATION (HORIZONTAL LINE).
SINCE THE POLAR STATIONS OVERESTIMATE, AND THE
EQUATORIAL STATIONS UNDERESTIMATE SU THIS RELATIONSHIP IS ASSOCIATED WITH THE REDISTRIBUTION OF THE
AVAILABLE THERMAL ENERGY OF THE ATMOSPHERE BY THE
GENERAL CIRCULATION.

FIG. 13 THIS RULE IS THE RELATIONSHIP BETWEEN THE
SURFACE UPWARD FLUX AND THE ATMOSPHERIC UPWARD
IR EMISSION (EMERGENT THERMAL RADIATION FROM THE
ATMOSPHERE ALONE): SU  2 EU . AGAIN WE HAVE A GOOD
AGREEMENT BETWEEN THE GLOBAL AVERAGES. THIS
RELATIONSHIP IS A DIRECT CONSEQUENCE OF THE GLOBAL
AVERAGE HYDROSTATIC EQUILIBRIUM STATE .

radiosonde observations from the NOAA Testing and
Evaluation Station are repeated. Many special atmospheric structures from different sources are evaluated
as well.
No atmospheric structures contradicting the above
rules are found. Even artificial profiles (like the
USST76 atmosphere) are consistent with the new flux
relationships. Judging from the correlation coefficients
(Figs. 10-13), none of these rules are perfect. In fact,
tight fits in these types of relationships are not

Development in Earth Science Volume 2, 2014

expected since the atmosphere is fundamentally a
stochastic medium. We must conclude that the above
rules represent the real radiative transfer properties of
the Earth-atmosphere system, and in order to get
closer to the cause of the greenhouse effect, one should
try to explain and understand all of them.
Although this study focused on IR fluxes at upper and
lower boundaries of the atmosphere, further results
are presented in Fig. 14 for the GAT vertical radiative
structure. Two sets of radiative fluxes (for the upper
and lower portions of the atmosphere) are plotted as a
function of the layer geometric thicknesses: z( z) 
ztop  z , and z ( z )  z , where

ztop  z70 is the 70 km top

altitude of the atmosphere. The lower boundary of the
whole air column is at z0  0 km. For some selected
altitudes numerical flux density data are presented in
Table 1.

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The detailed analysis of the vertical structure of the IR
radiation field will be the scope of another article.
Note, that the NOAA R1  A , and TA are practically
the same, but the lower atmospheric thermal structure
of the GAT is considerably colder as there is an
approximate 15 Wm-2 difference in SU at the ground.
This is a clear indication, that  A depends only upon
the real absorption properties of the atmosphere.
TABLE 1. VERTICAL RADIATIVE STRUCTURE OF THE GAT ATMOSPHERE.
ALTITUDES ARE IN KM, FLUXES ARE IN WM-2 , TRANSMITTANCES AND
OPTICAL THICKNESSES ARE DIMENSIONLESS. THE NOAA R1 61 YEAR
AVERAGE DATA ARE ALSO SHOWN IN RED COLOR.
DENSITY PROFILE AND

Altitude
60.0
50.0
30.0
15.0
10.0
5.00
2.00
0.00
0.00

B (z) IS THE FLUX

SU  BG  B (z  0) .

OLR
ED
EU
B(z)
202.1 0.2008 0.1793 202.1
253.4 0.7421 0.6553 253.6
154.9 4.774 5.620 153.5
115.1 17.52 20.98 111.4
152.7 52.82 48.73 160.1
212.3 149.0 108.6 262.4
239.6 237.6 156.1 331.9
251.8 309.9 193.2 379.7
256.4 321.5 195.4 395.0

ST
201.9
252.8
149.2
94.11
103.9
103.7
83.42
58.6
60.95

TA
τA
.9989 .0011
.9969 .0031
.9723 .0281
.8444 .1692
.6490 .4322
.3952 .9284
.2513 1.3810
.1542 1.8691
.1543 1.8688

In the next three figures (Figs. 15-17) the results of the
search for long term optical thickness trends in the 61
year long (1948-2008) NOAA NCEP/NCAR R1 reanalysis dataset are shown.

FIG. 14 GAT UPWARD AND DOWNWARD RADIATIVE FLUXES.
SOLID LINES ARE THE BOUNDARY FLUXES FROM LAYERS
BETWEEN ztop AND z ALTITUDES. BROKEN LINES ARE THE
BOUNDARY FLUXES FROM LAYERS BETWEEN 0.0 AND z
ALTITUDES. THE HORIZONTAL BLUE LINES ARE THE
ALTITUDES WHERE ED ( z )  EU ( z ) , AND SU ( z)  OLR( z) . AT THE
ALTITUDE OF THE BLACK CIRCLE ED ( z )  OLR( z ) .

The interesting features here are the approximate flux
equalities at some special altitudes: OLR( z2 )  ED ( z2 ) ,

OLR( z10 )  B( z10 ) , and ED ( z10 )  EU ( z10 ) . At the indicated levels the atmosphere has unique equilibrium
states which may largely affect the whole global
energy balance picture. For example, if the Kirchhoff
rule is valid for all altitudes, then the OLR( z2 )  ED ( z2 )
equation means that, at around the 2 km altitude, the
atmosphere above a cloud layer is in radiative
equilibrium. The last two equations imply that slightly
above the 10 km altitude the clear sky atmospheric
greenhouse effect stops, G  SU  OLR  0 .

Attempts to identify any significant changes in the
absorption characteristics of the atmosphere are
unsuccessful. For the above tasks HARTCODE is
pushed to extreme numerical accuracy, test runs for
small GHG perturbations were presented in M10. In
Fig. 15 the actual and expected atmospheric
absorption trends are compared for the full time
period. No change in the IR absorption is detected. In
Fig. 16 the results of the six sub-sets of the 61 year time
periods are also presented. The theoretical expectation
in each sub-set is met, no changes in the sample mean
optical thicknesses are apparent. However, both the
annual mean optical thickness and H2O column
amount are subject to random fluctuations.
Detailed Fourier analysis of the 61 year long time
series shows that a significant oscillation with a 3.529
year period is present in the data. This 'heartbeat' of
the atmosphere could be related to the ElNino LaNina cycles. Fig. 17 displays the observed changes
in  t A , t A , t S , H2O, and CO2 . The slight negative trend
in  t A and the large increase in CO2 concentrations is
certainly not consistent with the classic GE explanations, found in any textbook on climate change.

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Development in Earth Science Volume 2, 2014

Theoretical Interpretations
The analytical equations of the presented rules in Figs.
10-13 are summarized in the next four equations:

FIG. 15 ATMOSPHERIC ABSORPTION TRENDS IN THE LAST 61
YEARS. THE EXPECTED INCREASE IN THE ATMOSPHERIC
FLUX ABSORPTION DUE TO THE ~23 % CO2 INCREASE DURING
THIS TIME PERIOD IS NOT PRESENT.

ED  AA  SU A  SU (1  TA ) ,

(3)

OLR / SU  f ,

(4)

SU  3OLR / 2 ,

(5)

SU  2 EU .

(6)

In the range of the input data sets the atmospheric
Kirchhoff , radiative equilibrium, and virial rules, Eqs.
(3,4) appear to be valid for each individual sounding
and also for the global averages. The energy conservation rule and virial rule, Eqs. (5,6) are valid for only
the global averages. Although in M07 some successful
modelling and simulation results for the Martian
atmosphere have been published, the validity of Eqs.
(3-6) for other planets are not discussed here. Each
planet or moon in the solar system has its own distinct
physical condition and in each case the radiative
transfer problem must be formulated individually.
Atmospheric Kirchhoff Rule

FIG. 16 OPTICAL THICKNESS COMPUTATIONS FOR DIFFERENT
SUB-SETS OF THE NCEP/NCAR R1 ARCHIVE. SHORT TERM
FLUCTUATIONS ARE NOT RELATED TO CO2 INCREASE. THE
AVERAGE  A IN ANY TIME SERIES AGREED WITH THE
THEORETICAL EXPECTATION OF 1.87 .

Recently some researchers have raised the question of
the applicability of the Kirchhoff rule for atmospheric
radiative processes, see for example DeBruin (2010),
Spencer (2011). Since the atmospheric Kirchhoff rule
represents an empirical fact, such critiques do not have
much scientific ground. If a couple of hundred
atmospheric structures show the ED  AA approximate
equality, then the only way to refute this relationship
is to show an atmospheric structure which is violating
it. In a planetary radiative equilibrium situation the
strict ED  AA relationship at the lower boundary is
established with the material IR emissivity properties
of the ground surface and atmosphere.
The anisotropy in ED and the IR emissivity (or ref-

FIG. 17 THE OBSERVED GREENHOUSE TEMPERATURE
CHANGE IS NOT CONSISTENT WITH THE INCREASED CO2
CONCENTRATION. THE SURPRISING STABILITY OF t A
CANNOT BE EXPLAINED BY THE CLASSIC VIEW OF GREENHOUSE PHENOMENON, AND DOES NOT SUPPORT THE GE
BASED AGW HYPOTHESIS OF THE IPCC.

42

lectance) of the ground surface cannot be ignored. The
different forms of the monochromatic, flux, directional
etc., Kirchhoff laws are well known in the general
radiative transfer theory. It is also known that the
classic monochromatic Kirchhoff law is not valid in
the close vicinity of strong absorption/emission lines,
see M07. It is also difficult to adopt this law for
atmospheric IR flux densities where the inhomogeneous atmosphere is in permanent physical contact
with solid and liquid surfaces. The important finding
here is the ability of any real atmosphere to instantly
adjust its radiative structure to closely satisfy Eq. (3).

Development in Earth Science Volume 2, 2014

The physical explanation is very simple. The relaxation time of the IR radiation field is much smaller
than any macroscopic heat or energy transfer processes (related to the motion and thermodynamics) of
the atmosphere. The vibrational-rotational relaxation
time is in the order from 2×10-6 to 2×10-5 sec at 1 atm.
and 200K. The IR radiation field is close to quasi-static
equilibrium with the surrounding environment and it
instantly 'sees' the whole atmosphere, independently
of the dynamics of the system. The strict validity of the
spectral Kirchhoff law for a hypothetical isothermal
atmosphere is obvious. Such computation is presented
in Fig. 18. As can be concluded, the spectral Kirchhoff
law is exact, and perfectly reproduced, (see the red
line). Similar but spectral radiance simulations for
isothermal atmospheres are routinely performed to
test the numerical performance of LBL radiative
transfer codes, see Kratz et al. (2005). The computational (numerical) accuracy of our LBL code for flux
transmittance is excellent. The relative error in the
ED  AA equality is 100 (1  ED / AA )  2.2  10 6 % .
The conditions of the stability of the thermal structure
of an air column are also of interest. In Figs. 19 and 20
simulated global average flux transmittance, atmospheric downward emittance, and observed source
function profiles are presented for clear and cloudy
GAT atmospheres. In these simulations the cloud layer
is represented by a perfect black surface at a given
altitude with an infinitesimal vertical extension and in
thermal equilibrium with the surrounding air. The
thermal equilibrium and a perfectly black radiator are
also assumed at the ground surface, SU   B( z0 ) at zero
altitude. In Fig. 19 the whole atmospheric air column
is in radiative equilibrium with the surface air. This is
the obvious condition for the local thermodynamic
equilibrium (LTE), and the existence of a stable
temperature profile. At higher altitudes this figure
shows that any emitting cloud layer is also in radiative
equilibrium with the atmospheric column above. This
is also the condition of the LTE in the air column
above the cloud layer. We should note that in case the
global average atmosphere represents a long term
average structure which is in overall radiative balance
with the surrounding space, then the ~3 % global
average anisotropy effect in the Kirchhoff rule must be
accounted for by an effective spherical emissivity
factor of 1  0.9652 . In Fig. 20 the cloud layer is acting
as a cavity and the atmosphere below the cloud layer
is in radiative equilibrium with the emitting surfaces
at the upper and lower boundaries.

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FIG. 18 THE SPECTRAL KIRCHHOFF LAW IN ISOTHERMAL
ATMOSPHERE REQUIRES THE FOLLOWING EQUALITIES:

O LR  SU  ED / A  EU / A , AND EU / SU  ED / SU  A . THE SU  O LR / f
AND SU  ED / A EQUATIONS CANNOT BE SATISFIED SIMULTANEOUSLY, THEREFORE, SEMI-TRANSPARENT ISOTHERMAL
ATMOSPHERE CANNOT BE IN RADIATIVE EQUILIBRIUM.

FIG. 19 CLEAR SKY KIRCHHOFF LAW. THE ATMOSPHERIC
DOWNWARD EMITTANCE IS EQUAL TO THE ATMOSPHERIC
ABSORPTION OF THE SURFACE UPWARD RADIATION.

FIG. 20 CLOUDY SKY KIRCHHOFF LAW. UP TO ABOUT 3 KM
ALTITUDE THE MEAN ATMOSPHERIC EMITTANCE IS EQUAL
TO THE ABSORBED MEAN SURFACE RADIATION (FROM
GROUND AND CLOUD BOTTOM), TM IS THE WEIGHTED
AVERAGE FLUX TRANSMITTANCE.

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Development in Earth Science Volume 2, 2014

(see Appendix B, Eq. B8 in M07 ). However, the use of
radiation equilibrium terminology requires some
clarification. The definition of the piecewise radiative
equilibrium is given by B( )  (3/ 4 ) H  Bo , where

B( ) is the source function profile, H is the Eddington
flux, B0 is an integration constant, and  is the average
flux optical depth (measured from the TOA). In our
terminology, once a linear (actual or equivalent)
source function profile is established with the required
slope and B0 ( A ) , then the atmosphere is said to be in

FIG. 21 LOW LEVEL TEMPERATURE INVERSION AND
RADIATIVE EXCHANGE EQUILIBRIUM. HARTCODE
DETERMINED THE EQUILIBRIUM ALTITUDE USING THE
YELLOW DOTS FOR INTERPOLATION. THE ACCURACY OF THE
EQUILIBRIUM ALTITUDE IS ~ 4 M.

The question of the radiative exchange equilibrium
(introduced in M10) between the surface and a
particular part of the atmosphere is also studied. In
case thermal inversions are present in the temperature
profile, theoretically the surface must be in perfect
radiative exchange equilibrium with those atmospheric layers having the same temperature. For this
kind of computation 42 inversion cases are selected
from the TIGR2 data and the differences in the
absorbed and emitted radiations in each layer are
computed. Such kind of tests are very useful because
they can point to inconsistencies and programming
bugs in the computational algorithms.
In Fig. 21, results are presented for a cold and dry
arctic atmosphere showing ~5 K close-to-surface
temperature inversion. The return altitude of the
temperature profile (above the inversion layer) is
picked up with quite remarkable accuracy (~ 4 m). To
our satisfaction, HARTCODE computed the layer net
radiation according to our expectations. As we
mentioned already, the IR radiative imbalance at the
ground surface can easily be accounted for by
introducing material IR emission properties of the
atmosphere and the surface. The complete radiative
equilibrium at the surface can be, and must be
established.
Radiative Equilibrium Rule
The naming of “radiative equilibrium rule” is quite
straightforward. The new semi-transparent radiative
equilibrium equations, the derivation Eq. 4 from well
known principles of the general radiative transfer
theory are proved with sufficient mathematical rigor,

44

radiative equilibrium. In this case the atmosphere has
the required amount of GHGs (H2O, CO2, O3 etc.) to
support the SU  OLR / f relationship. An equivalent
form of Eq. (4) may be obtained using the EU 

ST  OLR defining identity: SU  EU / ( f  TA ) .
It is important to note, that at the theoretical derivation in M07 the gray approximation is just a simplified
terminology and applied only for the convenience of
dropping the wave number index in the equations. In
case of monochromatic radiative equilibrium, one may
rewrite the solution in the monochromatic form:
f  SU  OLR  , where f   f  ( A ) and  A are the monochromatic transfer function, and the monochromatic
flux optical thickness respectively. Integrating both
sides with respect to the wave number and applying
the mean value theorem of the calculus one may easily
arrive at Eq. (4).
In 2002 the only available theoretical relationship
between the IR optical thickness and the source function profile was the classic semi-infinite Eddington
solution, and its corrected versions (which tried to
resolve the surface temperature discontinuity problem). However, the related equations for semi-transparent atmospheres turned out to be mathematically
incorrect and should not be used. In a real global
average clear sky atmosphere where the net nonradiative energy fluxes equal to zero Eq. (4) holds
exactly.
The global average TIGR2 atmosphere used in this
article is quite close to the state of radiative
equilibrium. Test runs in the 0-120 km altitude range
show that SU  379.688 Wm-2, OLR  251.004 Wm-2,

f  0.661144 , and the error in Eq. (4) is negligible,
SU  OLR / f  0.037 Wm-2 . Because of the changes in

OLR and f , reducing the altitude range to 0-70 km, Eq.
(4) will overestimate SU by about ~1 Wm-2, but will
largely reduce the LBL computational burden.

Development in Earth Science Volume 2, 2014

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Energy Conservation Rule

Virial Rule

Before going into the details of the physical meanings
of the rules presented in Figs. 12 and 13, we should
spend some more time with the energy conservation
and virial rules. In these two rules the clear sky surface
upward flux is proportional with OLR and also, with
EU : SU  3 OLR / 2 and SU  2 EU . Unfortunately

Climate scientists tend to forget about the virial
theorem and they usually render it unusable for
climate research. The atmospheric virial rule, SU  2EU ,

these equations do not satisfy an obvious and necessary physical condition which is sometimes called the
transparent limit constraint.
For a transparent atmosphere,  A  0 , SU  ST  OLR ,
and ED  EU  0 conditions should be satisfied. To
implement the transparent limit constraint the
SV  ST / 2  ED /10 virial term is introduced. Adding

SV to the left hand side of Eq. (5), an equation which
obeys the transparent limit, and satisfies both of the
original equations is obtained. It is easy to show that
Eqs. (5) and (6) can be trivially satisfied with
ST / SU  1/ 6 : SU  2EU  2OLR  2ST  2(2SU / 3)  2ST ,
from which follows SU  6ST . The equivalent form of
this equation (using the observed ED  AA approximation) is ST  ED / 5  0 . It is assumed that the general
equation should be in the form of SU  SV  3 OLR / 2 ,
where SV  X (ST  5ED ) and X is a non-zero multiplier. In the transparent atmosphere limit SU  X SU 

3 SU / 2 , from which X  1/ 2 , and SV  ST / 2  ED /10 .
The final form is SU  ST / 2  ED /10  3OLR / 2 which can
be reshaped into a much simpler form:

SU  OLR /(3/ 5  2 TA / 5) .

(7)

From Eq. (7) and the definition of OLR the 5 / 3 
AA / EU simple relation and the 5/ 3  ED / EU approx-

shows a linear dependence between the surface upward flux density and atmospheric upward emittance.
Under hydrostatic balance the virial theorem relates
the potential energy and the internal energy. The virial
theorem may be expressed in different forms :
2T    0 , or 3( 1)U    0 , where T is the mean
kinetic energy,  is the gravitational potential energy,
 is the specific heat ratio and U is the internal energy
of the system, see Chandrasekhar (2010), Cox and
Giuli (1968) and Clausius (1870). In astrophysics the
Vogt-Russel theorem is a relationship between the
mass and luminosity of a star.
The above facts gave enough inspiration to try to
relate EU to the surface pressure or to the mass of the
atmosphere. The computations for the TIGR2 archive
are presented in Fig. 22. It is quite obvious that the
virial theorem is applicable for the Earth's atmosphere
and represents a permanent constraint on the IR radiation field. For the Earth atmosphere the differential
forms of the virial relationship is also confirmed quantitatively (not shwn here).
The critiques of the association of Eqs. (6) with the
virial concept in DeBruin (2010), Toth (2010), and in
the comments of other radiative transfer experts (see
E. Rabett, P. DeWitt, G. Schmidt, R. Pierrehumbert,
and B. Levenson in the Real Climate (2008), or Science
of Doom (2014) Blogs) have no theoretical and empirical foundations.

imation immediately follow. One should not forget,
that Eq. (7) (and its different forms) are applicable only
for global average atmospheres. For example, the
AA / EU ratios for the TIGR2 and USST76 atmospheres
are 1.666, and 1.766 subsequently, (see Fig. 3).
SU  3 OLR / 2 requires the validity of ED  AA . This
follows directly from the greenhouse identity, which
expresses the conservation of the radiant energy.
Applying the ED  AA approximation one arrives at the

SU  OLR  ED  EU equation, from which one may
readily obtain the SU  3 OLR / 2 relation. In complete
planetary radiative equilibrium ED  AA , and SU 

3 OLR / 2 . The violation of these rules leads to the
violation of the conservation of radiant energy as
explained in M07 Eq. (7) in page 7.

FIG. 22 VIRIAL CONCEPT IN HYDROSTATIC ATMOSPHERE.
INTERNAL ENERGY IS COMPUTED WITH ONE DEGREE OF
FREEDOM. GRAVITATIONAL POTENTIAL ENERGY DENSITY IS
REFERENCED TO THE SURFACE.

45

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Development in Earth Science Volume 2, 2014

Results and Discussion
As soon as sufficient confidence in the validity of the
individual clear sky atmospheric radiative transfer
rules is gained, the author is facing the interesting
problem of the empirically proven constant global
average clear sky flux optical thickness. Obviously the
energy conservation and radiative equilibrium rules
are the relevant equations which may be associated
with the overall planetary radiative balance and give
the needed theoretical support.
The simultaneous validity of Eqs. (4,7) requires the
solution of the OLR /(3/ 5  2TA / 5)  OLR / f transcendental equation which can be simplified into the
g  2 A / 5 form. The only unknown in these equations
is the equilibrium flux optical thickness which is the
sole function of the thermal and humidity structure of
the global average atmosphere.

FIG. 23 COMBINED TIGR 2 AND NOAA SIMULATIONS. THE ANNUAL MEAN NOAA R1 DATA NOT RESOLVED SUFFICIENTLY
TO SEE THE INDIVIDUAL SOUNDINGS. THE EMPIRICAL

 A   TA  1.87 RELATION SHIP IS FULLY SUPPORTED.

The numerical solution of the equation resulted in a
unique theoretical equilibrium flux optical thickness of
 AT  1.867561 . The other theoretical quantities are
derived from  AT : TAT  0.1545 , A T  0.8455 , f
g

T

 0.3382 . 

T
1

 0.9572 . Apparently this 

T
A

T

 0.6618 ,

does not

depend on any particular GHG concentration and it
might better be regarded as an invariant climate
parameter of the Earth-atmosphere system.
The first verification of the  A   AT equality is based
upon the TIGR2 and NOAA R1 radiosonde archives
(  A is the observed global average). The combined
results are summarized in Fig. 23. All annual global
mean optical thicknesses are practically equal to  AT
and supporting the  A   AT theoretical expectation. Fig.
23 shows also the TIGR2 EU / SU ratios (gray dots) and
the associated theoretical f  TA function (magenta
curve).
One should note that despite the relatively large
spread of the EU / SU dots, the global average
EU / SU  0.5089 ratio is consistent with the SU  2 EU

virial rule. In our atmosphere the individual EU / SU
ratios are also constrained by the TA , A , f , and g
radiative transfer functions (see the yellow shaded
area). The theoretical upper limit of  A is set by the

f  g  1/ 2 constraint:  Am a x  2.9475 . The 1  2 A / 5
(broken red curve) is a version of the energy conservation rule, and the green dot in the intercept of the f
and 1  2 A / 5 curves marks the position of  AT .

46

FIG. 24 STEADY-STATE CLEAR SKY CLIMATE MODEL WITH
CONSTANT IR OPTICAL THICKNESS. RED NUMBERS: OLR 1 ;
BLUE NUMBERS: SU  1 . IN RADIATIVE EQUILIBRIUM SU  FA / f ,
AND SU  FE (1   B ) / f , THEREFORE, SU MAY CHANGE ONLY
THROUGH FE OR  B .

The observed stability of the clear sky absorption
properties of the global average atmosphere may be
demonstrated with a heuristic clear sky RT model
presented in Fig. 24 (here F is the absorbed part of FA
within the atmosphere). This model is simplified in a
sense that the effects of the LW emissivity, reflectance,and anisotropy are ignored on the basis the
sustained radiative equilibrium requirement will
compensate all related imbalances. The net nonradiative processes K   K  are zeroed out on the basis,
that the planet is in radiative equilibrium and the
hydrological sub-system (or water cycle) is a closed
equilibrium process. The only requirement from the
model is the constant average IR optical thickness
which can be maintained around the  AT theoretical

Development in Earth Science Volume 2, 2014

value by the stochastic fluctuations of all flux density
ratios (around their respective planetary averages).
Compared to the real world relationships among the
boundary fluxes in Table 1, this model gives quite a
reasonable estimate of the normalized flux densities.
The constant long term global average clear sky flux
optical thickness does not leave much room for the
system to obey the energy conservation principle. It is
quite plausible to assume that the cloud cover is
responsible for simultaneously maintaining the radiative equilibrium and energy conservation requirements. Some aspects of the global average cloud cover
will be discussed in the next section, but the full
account of the detailed role of the hydrological cycle
and its quantitative effect on the flux density
components will be discussed elsewhere.
Radiative Equilibrium Cloud Cover
One of the most elusive problems of climate science is
the correct handling of the radiative effects of the
global average cloud cover. After decades of struggle
with the cloud forcing parameter and other mixed
physical quantities, the role of clouds in the climate
system remains hidden. It has been known for a long
time that the cloud cover follows the annual solar
cycle which is present in the SW energy input
( 330.25  FE  353.00 Wm-2), but a solid theoretical
foundation for the mechanism and the quantitative
C
methods for the practical evaluation of the  and h
parameters are not present in climate literature.

http://www.seipub.org/des

cover data, but unfortunately, large uncertainties in
A
the satellite OLR may result in any  within the

0.45    0.75 range. Because of the OLR A  OLR C
assumption and the limited capability of the
HARTCODE vertical layering routines (at that time),
C
the accuracy of our  and h was unknown. It was
impossible to prove the consequential AAC (h C )  OLR ,
  f ( AT ) , and ED (h C )  OLR C relationships from the

Kirchhoff rule (see page 19 in M07).
In the recent approach it is assumed that the GAT
atmosphere represents the global average structure
reasonably well, and the F0 , (and consequently FE ),
are also known with sufficient accuracy. One may
construct two discrete sets of data (  A ( B , h C ) , and
 E ( B , h C ) ) from LBL simulations of the OLR , OLR ,
A

C

and OLR flux densities:


A

 ((1   B ) S E  OLR ) /(OLR C (h C )  OLR ) ,

 E  (OLR A /(1   B )  SU ) /( SUC ( h C )  SU ) .

(8)
(9)

 A ( B , h C ) and  E ( B , h C ) are the cloud fractions

from

the  A OLR C  (1   A ) OLR  FA and  E SUC 

(1   E ) SU  FE equations, respectively. Note that in

spherical geometry the cloud fraction does not depend
on the altitude. In the two-dimensional optimization
problem, only one global average cloud layer is
assumed and the ||  A   E || 2 norm is minimized. For
obtaining accurate  , and  B , the vertical resolution of
the HARTCODE altitude vector is set to 40 cm.

Accurate RT computations using HARTCODE lead to
the discovery of four fundamental atmospheric
radiative transfer rules. As an application, in this
C
section the radiative equilibrium  , and h which are
consistent with the above rules and the related
constraints are determined. In view of the  A   AT and
OLR  SU f ( TA ) clear sky LW radiative equilibrium

requirements it is obvious, that the task of assuring the
all-sky radiative balance and the OLR A  FE  FR top
level energy conservation constraint, is left entirely to
C
the  , and h parameters.
We should mention, that in M07 an attempt is already
C
made to compute  , and h from the atmospheric
Kirchhoff, and the energy conservation rules:  
( SU  3 OLR A / 2) / ( SU  SUC ) , where SU  382 Wm-2,
C
OLR A  235 Wm-2, and SU  333 Wm-2. The resulted

  0.6 and the related h  2.05 km cloud top altiC

FIG. 25 RESULTS OF THE MULTIVARIABLE NONLINEAR
OPTIMIZATION. THE EQUILIBRIUM CLOUD COVER, BOND
ALBEDO, AND CLOUD TOP ALTITUDE ARE  T  f ( AT )  0.6618 ,

 B  GE  0.3013 , AND h C  1.9160 KM.

tude fit well into the wide range of published cloud

47

http://www.seipub.org/des

Development in Earth Science Volume 2, 2014

The range of  B is not that critical, here the solution
somewhere between 0.294   B  0.306 is expected. In
Fig. 25 the three-dimensional view of the optimization results are shown. In the close vicinity of the
minimum, the norm changes 2-3 orders of magnitude,
indicating a very sharp extremum. The results show
extremely good numerical agreement between  T
and f T :  T  f ( AT )  0.6618  107 .
The equilibrium albedo and cloud top altitude are:
 B  0.301290611 , and h C  1.9160 km, respectively.
C
Some other ways of finding the accurate  , and h

have been presented in Miskolczi (2014).
The independent empirical global average cloud cover
estimates from the ISCCP are consistent with our
results. From a 20 year long time series (ISCCP−D2
198307−200806 in Van Andel (2010)) a global mean 

FIG. 26 FLUX DENSITY SPECTRA OF THE ALL-SKY GAT ATMOSPHERE. THE EQUIVALENT BLACKBODY SPECTRA B(t A ) , AND

B(tS ) ARE EQUAL TO THE EQUIVALENT BLACKBODY SPECTRA
OF B(t ANASA ) , AND B(tSNASA ) .

of 66.38 +/− 1.48 % was reported. The 10 year average
ISCCP data show similar global cloud cover,   66 %,
Ollila (2013).
According to the EDA  g ED  f OLR C d relationship the
Cd

LW back radiation (through the OLR term) depends
on the cloud altitude. Using our equilibrium transfer
function f T and h C the back radiation is EDA  345.98
Wm-2. This value is quite consistent with the observed
EDA  345.4 Wm-2 (ISCCP-FD value for the CERES
period from March 2000 to May 2004 ), see Table 2 in
TFK09, and the EDA  345.6  9 Wm-2 quoted by S12 in
their NATURE article. W13 gives the best estimate of
the back radiation as EDA  342  5 Wm-2.

FIG. 27 SPECTRAL ALL-SKY GREENHOUSE EFFECT GA . FR AND
THE EFFECTIVE GAe AGREE REASONABLY WELL.

The spectral distributions of the all-sky fluxes of the
GAT atmosphere are presented in Fig. 26. The all-sky
spectral GE shown in Fig. 27. The numerical values of
the integrated fluxes show that the GAT atmosphere is
practically a radiative equilibrium structure.
Evidently, the all-sky greenhouse effect locked to the
reflected solar radiation: G A  103.0418 Wm-2, and

FR  103.032 Wm-2. The clear sky g locked tightly to
 TA : (SU  OLR) / SU  0.33684 , g ( TA )  0.3382 . In Fig. 26
superscripts are references to the NASA planetary fact
sheets, NASA GSFC NSSDC (2012). For reference, in
Table 2 the detailed numerical results of the
equilibrium flux density components are shown. The
Kirchhoff rule seems to be perfectly satisfied. The
AAC  OLR and EDC  OLR C equalities put the full control of the planetary radiative equilibrium into the
hand of the global average cloud cover.

48

FIG. 28 SOLAR AND TERRESTRIAL EQUILIBRIUM BLACKBODY
SPECTRA. THE LIGHT BLUE CURVE IS THE SOLAR SPECTRUM,
A
THE DARK BLUE IS THE OBSERVED LW SPECTRAL OLR FROM
THE GAT ATMOSPHERE. THE CYAN DOT MARKS THE MAXIMUM OF THE 273.15 K BLACKBODY SPECTRUM.

Development in Earth Science Volume 2, 2014

http://www.seipub.org/des

TABLE 2. GAT HIGH ACCURACY BOUNDARY FLUXES IN WM-2 . REGION
BOUNDARIES ARE IN KM. IN CLOUDY ATMOSPHERE THE
AND

AAC  OLR (RED)

EDC  OLR C (GREEN) EQUALITIES ARE ONLY SATISFIED AT A SINGLE

h C  1.916
REGION
0-70
1.92-70

  (G   B FE ) / (G  G C ) and  B  (G   (G  G C )) / FE

KM ALTITUDE.

UPWARD

three phases permanently present in the atmosphere
and two phases on the ground surface, the stability of
the planetary climate will be controlled by the

DOWNWARD

SU

AA

OLR

ED

379.69
333.82

321.12
251.12

251.79
240.14

309.93
240.15

Considering the temporal and areal variability in the
local water vapor content of an air column, one has to
admit that the Earth's atmosphere possesses enormous stability against fluctuations in its global average flux optical thickness. In our understanding, the
source of this stability is related to two natural causes.
One is the favourable orbital parameters of the Earth,
and the other is the permanent presence of the three
phases of the H2O in the boundary layer. According to
the Maxwell rule, the system as a whole has zero
thermodynamic degree of freedom, the phase temperature of the system must be the triple point of the H 2O
(273.16 K). In Fig. 28 we demonstrate, that the
maximum of the all-sky thermal emission spectrum of
the planet is, in fact, a spectral distribution of maximum radiation entropy.
Conclusions
In this research the IR radiative processes in the
climate system are studied quantitatively. Observed
empirical facts point to the existence of a climate
invariant constant global average clear sky flux optical
thickness of  A  1.87 . Theoretical support has also
been established with four fundamental radiative
transfer relationships and a theoretical  TA  1.8676 flux
optical thickness. The clear sky OLR  251.79 Wm-2
and SU  379.69 Wm-2 fluxes are fully consistent with
the OLR  SU f ( TA ) , and the OLR  SU (0.6  0.4 TAT )
theoretical requirements. It is also shown that the
global average atmosphere with its effective cloud
layer at h C  1.9160 km and a geometric cloud fraction
of   f ( TA )  0.6618 is in radiative balance with the

FE  341.97 Wm-2 TOA available solar radiation. It has
been proven quantitatively that the conservation of
radiant energy is established by the  B  g A  0.3013
and   f ( TA )  0.6618 equalities. In this respect the
two equations linking the Bond albedo to the cloud
cover and the all-sky normalized greenhouse factor
have fundamental importance. As long as the Earth
has unlimited water supply (in the oceans) with its

equations. These two equations, together with the
Clausius-Clapeyron equation, will regulate the
transfer of the latent heat through the boundary layer
in such a way that the net amount maintains the
planetary radiative balance. In this regard the thermodynamic boundary layer may be defined as the
combined surfaces where the different phases of the
water are in direct physical contact with each other
and with the surrounding material.
The apparent role of the Clausius-Clapeyron equation
is to convert temperature differences to radiative
fluxes (to and back), and by doing so to assure that no
temperature-radiation feedback exists in the system.
The only solution to the Earth's ground surface
temperature is tG  (SU /  / 1 )1/4  288.6  0.1 K. The

tP  (tM  11/4tG ) / 2 phase temperature is 273.17  0.1 K,
where tM   tM4   1/ 3  260.29 K is a unique universal
temperature. The empirically established climatological normalized GF of gmA  0.4 is also reproduced
well and proved by the gmA 1 OLR A /  tG4  0.3992
equation.
Of course the whole dynamically controlled system
has no real instantaneous equilibrium state. However,
the radial (or mass) oscillation of the system will be
able to handle the energy conservation and energy
minimum principles as required by the time constants
of the different latent heat reservoirs. In summary, the
complex task of the relatively fast responding global
mean cloud cover is to assure the conservation of
radiant energy and momentum on a global scale,
maximize the LW cooling to space (radiative
equilibrium), while observing the thermodynamic
constraints applicable to large heterogeneous systems
(Maxwell rule).
The quantitative proof of the radiative equilibrium
state of the Earth-atmosphere system is alone a
remarkable achievement of planetary science. The
proposition here is to consider the global average
cloud cover as the only component of the climate
system, which is able to respond to and regulate the
planetary radiation budget in a relatively short time.
The greenhouse effect of the Earth's atmosphere is a
global scale equilibrium process which rests on the
chaotic nature of the humidity field and the stability of

49

http://www.seipub.org/des

the total atmospheric mass. Consequently, none of any
local or regional weather phenomenon is related
directly to its magnitude and tendency.
Unfortunately the Nobel Laureate IPCC is not a scientific authority, and their claim of the consensus and
the settled greenhouse science is meaningless. The
quantitative results of this paper massively contradict
the CO2 greenhouse effect based AGW hypothesis of
IPCC.
In our view the greenhouse phenomenon, as it was
postulated by J. Fourier (1824), estimated by S.
Arrhenius (1906), first quantified by S. Manabe and R.
Wetherald (1967), explained by R. Lindzen (2007), and
endorsed by the National Academy of Science and the
Royal Society (2014), simple does not exist.
However, research must continue to find and establish
the real causes and the true trends in global temperature change that may be present behind the natural
fluctuations. The greenhouse science is not settled, the
presented results warrant further efforts to investigate
many detailes of the surface radiative equilibrium
processes.
ACKNOWLEDGEMENTS

I am very grateful for the help and support obtained
from K. Gregory, and C. Game. Without their computational assistance, this research project could not exist.
I also wish to thank K. Vinnikov, A. Rörsch, D. Hagen,
S. Welcenbach, C. Wiese, G. Fulks, D. Brooks, W.
Guang, Y. Shaomin, L. Szarka, V. Wesztergom, Sz.
Barcza, Z. Kollath, Z. Toth, S. Kenyeres and P. Nemeth,
for their continuous attention and valuable advice. The
help and scientific contributions from my two friends,
N. VanAndel and J. Pompe, who both died while in
the midst of this work should also be remembered.
Credit also go to C. Dubay, E. Carbone and J. Ginsberg
for their help with the technical editing of the
manuscript. The work of the reviewers and the editors
of the journal is highly appreciated.

Development in Earth Science Volume 2, 2014

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Ferenc Miskolczi Born in Budapest,
Hungary in 1947. At the Eötvös Loránd
University, Budapest, he earned an MS
degree in nuclear physics (1971), and a
PhD degree in astrophysics (1975). At
the Hungarian Academy of Sciences he
earned another PhD in Earth Sciences
(1981). He also holds a diploma in highlevel computer programming.
He specialized in the experimental and theoretical aspects of
infrared atmospheric radiative transfer. During the 80s, and
90s, he taught and did research at several universities. Before
his international career, he was the head of the Department
of Atmospheric Radiation at the Institute of Atmospheric
Physics, Budapest (Hungary). He was a research associate in

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the International Centre for Theoretical Physics, Trieste
(Italy), a research scientist of the University of Maryland
(USA), BSRN Staion Scientist in Ilorin (Nigeria), and a senior
principal scientist at the Raytheon STX Corporation (USA).
He worked as Co-Principal Inveestigator on several NASA
projects related to the problems of atmospheric remote
sensing and planetary radiation budget. In 2003 he discovered and established the equations governing the
infrared radiative transfer in semi-transparent atmospheres.
In 2006 he resigned from NASA in protest due to unresolved
publication issues related to his AGW related results.
Recently his research interest has been the planetary greenhouse effect. His new idea of the greenhouse effect is getting
more and more widely recognized. He is Foreign Associate
Member of the Hungarian Academy of Sciences.