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Article
Relative importance of carbon dioxide and water in the greenhouse effect:
Does the tail wag the dog?
Demetris Koutsoyiannis
Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical
University of Athens, Zographou, Greece, dk@itia.ntua.gr, http://www.itia.ntua.gr/dk
Abstract: Using a detailed atmospheric radiative transfer model, we derive macroscopic relation-
ships of downwelling and outgoing longwave radiation, useful for hydrological practice. We vali-
date them using empirical formulae based on downwelling radiation data, which are in common
use in hydrology, as well as satellite data for the outgoing radiation. We use the macroscopic rela-
tionships to infer the relative importance of carbon dioxide and water vapour in the greenhouse
effect. The results show that the contribution of the former is 4% – 5%, while water and clouds dom-
inate with a contribution of 87% – 95%. The minor effect of carbon dioxide is also confirmed by the
small, non-discernible effect of the recent escalation of atmospheric CO2 concentration from 300 to
420 ppm, which is quantified at 0.5% for both downwelling and outgoing radiation. Water and
clouds also perform other important functions in climate, such as regulating the heat storage and
the albedo, as well as cooling the Earth’s surface though latent heat transfer, with a contribution of
50%. By confirming the major role of water on climate, these results suggest that hydrology should
have a more prominent and more active role in climate research.
Keywords: greenhouse effect; longwave radiation; water vapour; carbon dioxide; evaporation
A good rule of thumb to keep in mind is that anything that calls itself 'science' probably isn't.
J.R. Searle [
1
]
1. Introduction
A notable feature of the current period is that the classical value of science as the
pursuit of the truth, independently of other interests, is gradually being abandoned [
2
].
People pride themselves on being scientists and (political) activists at the same time [
3
],
while calls for political actions to “save the planet”, including enhanced global governance
mechanisms, are published even in scientific journals (e.g. [
4
]). A relevant example is pro-
vided by the high-profile journal Nature, which declares that it is “committed to support-
ing the research enterprise” [
5
], but admits being involved in politics and proclaims that
this is the right thing [
6
,
7
], even though it is recognized that this affects scientific credibil-
ity and causes loss of confidence in science [
8
,
9
]. It is thus natural to wonder if articles
published in scientific journals promote the truth or political aims, perhaps hiding or dis-
torting the truth, and even reversing cause and effect [
10
-
13
].
Seeking truth may have been even more difficult due to a recent development, the
appearance of artificial intelligence (AI) bots, which certainly affects common opinion, as
well as scientific perception and advancement. Assuming that AI bots more or less reflect
the common perception in the scientific community and the general public, the Microsoft
Bing’s Chat bot was asked the following question:
Which gas determines the greenhouse effect on Earth?
The bot provided the following answer (also supporting it with references [
14
-
16
]):
The greenhouse effect is the process through which heat is trapped near Earth’s surface by
substances known as ‘greenhouse gases.’ These gases consist of carbon dioxide, methane,
2
ozone, nitrous oxide, chlorofluorocarbons, and water vapor. Water vapor, which reacts to tem-
perature changes, is referred to as a ‘feedback’, because it amplifies the effect of forces that
initially caused the warming. Scientists have determined that carbon dioxide plays a crucial
role in maintaining the stability of Earth’s atmosphere. If carbon dioxide were removed, the
terrestrial greenhouse effect would collapse, and Earth’s surface temperature would drop sig-
nificantly, by approximately 33°C (59°F).
A literature search reveals that the AI bot essentially (and selectively) reproduces the
following statements by Lacis et al. [
17
]:
Noncondensing greenhouse gases, which account for 25% of the total terrestrial greenhouse
effect, thus serve to provide the stable temperature structure that sustains the current levels
of atmospheric water vapor and clouds via feedback processes that account for the remaining
75% of the greenhouse effect. Without the radiative forcing supplied by CO₂ and the other
noncondensing greenhouse gases, the terrestrial greenhouse would collapse, plunging the
global climate into an icebound Earth state. […]
If the global atmospheric temperatures were to fall to as low as TS = TE [where TE = 255 K is
the global mean effective temperature] the Clausius-Clapeyron relation would imply that
the sustainable amount of atmospheric water vapor would become less than 10% of the current
atmospheric value.
On the other hand, Koutsoyiannis and Vournas [
18
] recently examined longwave ra-
diation observations extending over a period of 100 years and found that the observed
increase of the atmospheric dioxide concentration ([CO₂]; from 300 to 420 ppm) has not
altered, in a discernible manner, the greenhouse effect, which remains dominated by the
quantity of water vapour in the atmosphere. Naturally, given the uproar about the en-
hancement of greenhouse effect due to human emissions by fossil fuel combustion, this
finding appeared surprising to many. Some (including a knowledgeable reviewer of [18])
postulated that this would be expected for the downwelling longwave (LW) radiation
flux, which was the subject of Koutsoyiannis and Vournas [18], but would not be the case
for the outgoing radiation, where the effect of [CO₂] increase would be substantial. How-
ever, no long data series exist to verify such a conjecture and hence this question was not
investigated in [18], whose scope was to make inference based on data.
Hence the following research questions are raised:
1. Are Lacis et al. (and the bot) right about the importance of CO₂ in the greenhouse
effect, and is it meaningful to state that without it the terrestrial greenhouse
would collapse? Or is the effect of CO₂ negligible as Koutsoyiannis and Vournas
[18] claimed, and that of H₂O dominant?
2. Is the role of H₂O as a greenhouse gas limited to the downwelling LW flux or does
it extend also to the outgoing LW flux?
We will examine these questions below, noting that the first one refers to a fictitious
case (removal of atmospheric CO₂) for which no empirical data can exist. Rather, paleocli-
matic studies and geological facts suggest that CO₂ existed, mostly in much higher con-
centrations than today, in most periods of Earth’s history, and also before oxygen ap-
peared in the atmosphere. Also, the second question cannot be studied on an empirical
basis, as no long-term data exist (systematic satellite measurements of outgoing LW flux
have only been made in the 21st century). Therefore, to study these questions we need to
resort to theoretical arguments and analyses. We will do this by applying the established
greenhouse theory and by enrolling standard models, without considering doubts that
have been cast on the validity of the theory or alternative hypotheses (e.g. [
19
,
20
]).
Human CO₂ emissions are 4% of the total [13] but there are also human H₂O emis-
sions over the terrestrial part of Earth of a comparable percentage [
21
,
22
]. Specifically,
according to Koutsoyiannis [
23
] the quantity of evaporation and transpiration over land
is 91 400 km3/year. According to Food and Agriculture Organization of the United Nations
[
24
], the human water withdrawal in 2010, including the evaporation from reservoirs, was
4.3 km3/year, of which 69% and 19% were for agricultural and industrial use, respectively.
3
Considering the fact that almost all of agricultural and a large part of industrial water are
evaporated, as well as the increasing trend in water withdrawal, with simple calculations
we may conclude that the current human addition to natural water cycle over land is
about 4%. One could speculate that each of these 4% additions might influence climate to
a degree comparable to that percentage, but the reasons that only the influence of CO₂ is
investigated and highlighted by the scientific community, being regarded a driver of cli-
mate, despite the fact that H₂O is much stronger as a greenhouse gas, should be non-sci-
entific.
We may note, though, that this percentage becomes lower, of the order of 1%, if we
also consider the evaporation over oceans. However, it appears reasonable for this esti-
mate of human water emissions to consider only the terrestrial part of Earth, because of
the high local variability and the small residence time of atmospheric water. Indeed, the
mean residence time of atmospheric water is a few days (specifically, 12 250 km3 / (522 700
km3/year) = 0.023 years = 8.6 d, where the total volume of liquid water in the atmosphere,
12 250 km3, was taken as the average atmospheric water content, 24.0 kg/m3, as estimated
from the ERA5 Reanalysis over the globe for the period 1950-2023, while the atmospheric
inflow volume of water, 522 700 km3/year, was taken from [23]). In contrast, CO₂ is well
mixed as it has a much higher residence time, of a few years (specifically, 870 Gt / (226.9
Gt/year) = 3.8 years, where the total mass of atmospheric CO₂, 870 Gt was taken from the
Intergovernmental Panel on Climate Change (IPCC) report [
25
], namely its figure 5.12,
while the atmospheric inflow mass, 226.9 Gt/year, was taken from [13]).
The thesis expressed in this paper is that none of these anthropogenic additions to
hydrological and carbon cycle drives climate. On the other hand, both H₂O and CO₂ are
important elements of climate and their quantities and fluxes are determined by natural
processes, with the human factor being rather negligible. Both are elixirs of life and in this
respect they act complementary to each other. Thus, it may be pointless to compare them
to each other. Yet this comparison is the main focus of this paper, as lately the scientific
efforts to study each of them has been inversely proportional to their respective im-
portance.
By its definition by UNESCO [
26
], hydrology is the science which deals with the wa-
ters of the Earth, and its domain covers the entire history of the cycle of water on the Earth.
Water is a critical element of life and of climate as well. While climate has become a hot
topic and its research a top priority, it is odd that hydrology has lost importance, as evi-
dent from the abundance of papers examining climate change impacts and applying
model projections for the future based on CO₂ emission scenarios. This totally misses the
fact that water is the dominant element on Earth in driving climate and that the hydrolog-
ical cycle is self-ruling rather than a feedback or impact of another cycle—namely the car-
bon cycle, which has also been downgraded to an issue governed by human carbon emis-
sions (the 4% of the total).
By emphasizing the relative importance of water in climate, in comparison to carbon
dioxide, this paper tries to show that the picture of Earth’s climatic system may have been
distorted, and to give prominence to hydrology (and its branch of hydrometeorology) in
climate. The paper is made as a stand-alone and therefore it includes a synopsis of the
related theoretical concepts and a model (Section 2). Its foundation is not only theoretical,
but also empirical, utilizing observed data (Section 3). By combining the model and data,
it extracts simple macroscopical empirical relationships representing the greenhouse ef-
fect as accurately as the detailed model whose results are based upon (Section 4). These
relationships are tested against observational data (Section 5) and their simple and ana-
lytical expressions, which enable extraction of partial derivatives, allow the comparison
of the effect of water relative to other greenhouse gases (Section 6). The findings are put
in a more general context (Section 7) and allow drawing relevant conclusions (Section 8).
4
2. Theoretical Background
2.1 Quantification of Greenhouse Gases
The typical quantification of the abundance of a specific gas in a gas mixture is
given by its concentration, defined in terms of mole fraction as:
(1)
where and are the numbers of moles of the gas contained in a specified volume
and the total amount of moles of all constituents in the same volume, and and are
the respective number of molecules; note that , where = 6.022×1023 mol–1 is
the Avogadro constant (and likewise for ). Another measure is provided by the mass
fraction:
(2)
where and are the mass of the gas and the total mass of all constituents at a spec-
ified volume , and are the respective densities (, and
and are respective molar masses (in units of mass per mole).
The partial pressure of , , provides another quantification. By virtue of the ideal
gas law and the above equation we find:
(3)
where 8.314 J K–1 mol–1 is the universal gas constant and T is the temperature.
As an example, we consider the atmosphere in standard conditions at mean sea level,
i.e., 1013.25 hPa, T = 15 °C = 288.15 K, 1.225 kg/m3. The molar mass of the
mixture is 28.96 kg/kmol. Considering CO₂ as the specific gas of inter-
est, with molar mass 44.01 kg/kmol, at concentration [CO₂] = 400 ppm, we find
that
607.8 ppm and 0.4 hPa.
For the water vapour in the atmosphere, whose partial pressure by convention is
denoted as , the concentration varies substantially in space and time. The mass fraction,
known as specific humidity, is found, after algebraic manipulations, to be:
(4)
where ε is the ratio of the molecular mass of water to that of the mixture of gases in the
dry air, i.e., ε = 18.016/28.966 = 0.622. As an example, for a typical value of 15 hPa (see
Section 6.2) and standard atmospheric conditions as above, we find q = 9.3‰ and [H₂O] =
14.9‰ (almost 40 times higher than in the above typical example of CO₂).
The water vapour pressure has a thermodynamic upper limit, the saturation water
vapour pressure, which is a function of the temperature, T:
(5)
where (T0, e0) are the coordinates of the triple point of water (
), is the specific gas constant of water vapour (
), is the specific heat at constant pressure of the water vapour (
), is the specific heat of the liquid water , and
, with k the Boltzmann’s constant, and ξ is the amount of energy required
for a molecule to move from the liquid to gaseous phase. The parameter is related to the
latent heat of vaporization, Λ:
(6)
5
which is valid for any , where at the triple point . By substitution
of the various constants in (5), the following form of the equation is derived (first found
in [
27
]):
(7)
where the constant
was slightly modified in its last two decimal digits to optimize
the fit to the data. This is the celebrated Clausius-Clapeyron equation, which was recently
rederived in a pure stochastic context by maximizing the entropy, i.e., the uncertainty, in
a single water molecule [
28
,
29
]. Notably, the maximization of uncertainty at the micro-
scopic level yields a law that at the macroscopic level is nearly deterministic.
For completeness, we also modify the equation for the saturation water pressure over
ice. In this case it suffices to replace in equation (7) the specific heat of liquid water with
that of ice, , and the latent heat of vaporization with that of sublimation, resulting in a
constant to substitute for . Following Ambaum [
30
], we adopt the value
and hence Optimizing the average relative square er-
ror from benchmark values provided by Murphy and Koop ([
31
] Appendix C) for tem-
peratures 150 to 273.16 K, we find
. Hence, the equation for the saturation
water pressure over ice becomes:
(8)
A state in which the vapour pressure is lower than the saturation pressure is
characterized by the relative humidity:
(9)
which serves as a formal definition of both the relative humidity and the dew point .
2.2 Quantification of Longwave Radiation
The longwave radiation flux, , from a body (measured as energy per unit time and
unit area, typically ) at temperature (measured in kelvins) is described by the
Stefan-Boltzmann law:
(10)
where the Stefan–Boltzmann constant, , and the emissivity
of the body (dimensionless). For a black body radiator, . The Stefan–Boltzmann con-
stant is a fixed physical constant as it is related to other fundamental physical and math-
ematical constants by
(11)
where π is the ratio of a circle’s circumference to its diameter, is the Boltzmann’s con-
stant, is the Planck’s constant, and is the speed of light in vacuum.
At the Earth’s surface, the three LW radiation fluxes of interest are: (a) , emitted by
the surface (solid or liquid) directed upward, (b) , emitted by the atmosphere directed
downward, and (c) , the net emission. These are given as:
(12)
where in the last equation a minor term of reflected upward longwave radiation has been
neglected. The temperature of the surface, , is well defined and the emissivity is close
to 1, usually taken . However, in the atmosphere the temperature varies substan-
tially and the quantity is the integration of the radiation process across the entire at-
mosphere.
6
The theoretical basis for such integration is described by Goody [
32
]. Based on this
theoretical basis and some assumptions on the atmospheric profiles (nearly standard at-
mosphere), Brutsaert [
33
]was able to express analytically (by integration) the atmospheric
radiation near the surface for clear sky, and eventually find the effective emissivity as
a function of the atmospheric temperature, ,
taken at a level near Earth’s surface, and
the partial pressure of water vapour, , taken at the same level:
(13)
He also proposed a simplification by fixing to the average Earth’s temperature
near the surface, i.e. to 288 K, whence Equation (13) becomes
(14)
A modification of the Brutsaert’s Equation (13) was proposed by Prata [
34
], which is
expressed as:
(15)
with w representing the atmospheric water content (most commonly known with the mis-
nomer ‘precipitable water’), found by linear regression on radiosonde data and expressed
in cm. We may observe that for, Brutsaert’s equation (13) results in zero emissivity,
while Prata’s Equation (15) has a nonzero minimum of and, in this way, it de-
scribes the non-condensing greenhouse gas contribution to emissivity.
Decades earlier, empirical relationships of the same type (and with nonzero mini-
mum) had been proposed, among which the earliest, most celebrated and most popular
is that by Brunt [
35
,
36
]:
(16)
Furthermore, Brunt [36] using several data sets, he fitted the mathematical expression
(17)
and found different values of the coefficients for each data set. An average fitting
for all cases is:
(18)
With Penman’s [
37
] celebrated paper on natural evaporation, this quantification be-
came an essential part of hydrological practice in calculating evaporation. Essentially,
Penman used Brunt’s Equation (18), also assuming that and . Indeed, it can
be readily seen that Penman’s original equation (numbered (7) in his paper [37]), which
for clear sky conditions reads
(19)
is a direct result of Brunt’s Equation (18) and these assumptions, even though Penman
did not make a distinction of the two components seen in equation (12)
Later Penman’s equation was complemented by Monteith [
38
] to estimate water re-
quirements of crops, thus shaping what has been called the Penman-Monteith method.
This became a standard of the Food and Industry Organization (FAO), initially in the ver-
sion by Doorenbos and Pruitt [
39
] and later in the version by Allen et al. [
40
]. In both
versions, the downwelling longwave radiation (again assuming that and ) is
calculated as
(20)
7
Subsequently, a large variety of similar empirical relationships were proposed by
several researchers, critical reviews of which can be found in Carmona et al. [
41
, Guo et
al. [
42
] and Wong et al. [
43
], to mention the most recent.
Essentially, these simple equations quantify the greenhouse effect due to the presence
of water vapour in the atmosphere. They are used to calculate evapotranspiration, which
represents a substantial component of the hydrological balance—and also the most intrac-
table and difficult to measure. At the same time, evaporation calculations are most essen-
tial for agricultural irrigation practice. It is impressive that they quantify the LW radiation
flux based on a single variable, , in addition to the temperature, , and a single value
thereof, that at Earth’s surface. This clearly reflects the fact that it is the water vapour that
determines the greenhouse effect on Earth's atmosphere, contrary to the public perception
(and the scientific attempts to back it, as seen in the Introduction) that this is CO₂.
Yet it is useful to quantify the contribution of CO₂ to the greenhouse effect and com-
pare it to that of water vapour. In this, the above formulae do not help, and we need to
enrol detailed modelling of the spectroscopic properties of the atmosphere. Also, while
formulae are useful for the downwelling radiation, they do not quantify the outgoing ra-
diation at the top of the atmosphere (TOA). For those tasks we may use detailed models
as described in the next subsection.
2.3 Detailed Spectroscopic Modelling of the Atmosphere
There are several databases of spectroscopic parameters and codes that perform de-
tailed modelling of radiation in the atmosphere. Of these, here we use the MODerate res-
olution atmospheric TRANsmission model, or MODTRAN [
44
-
46
], which has been exten-
sively validated in its over 30-year history, and serves as a community standard atmos-
pheric band model. It simulates the emission and absorption of infrared radiation in the
atmosphere and, in particular, the effect of wavelength-selective greenhouse gases on
Earth's LW radiation flux.
The specific implementation used is that of the University of Chicago, readily pro-
vided as an interactive web application [
47
,
48
]. The application is based on five different
locality cases, which differ most significantly in their temperature, water vapour and
ozone profiles, as specified in the Appendix A. The application offers default values of
several characteristics for each locality profile, but also allows modifying these values
(e.g., by offsetting temperature and holding fixed either the water vapour pressure or the
relative humidity, or by multiplying the water vapour scale by a constant). Finally, in ad-
dition to the clear-sky conditions, it enables using a number of different cloud types, as
will be described in Section 4.2.
A typical graphical output of the MODTRAN application is seen in Figure 1. This is
for the standard tropical atmospheric profile, also in comparison to the case that the stand-
ard [CO₂] of 400 ppm is replaced by that of 800 ppm (the difference is difficult to discern).
8
Figure 1 Output of the MODTRAN model for the standard tropical atmospheric profile, in compar-
ison to the case that the standard [CO₂] of 400 ppm is replaced by that of 800 ppm. The outgoing
radiation corresponds to the level of 100 km above the ground. The ground temperature is 299.7 K
(26.55 °C) and the total outgoing LW flux is 298.52 W/m² for [CO₂] = 400 ppm and 295.129 W/m² for
[CO₂] = 800 ppm (difference –3.36 W/m²). Graph generated from https://climatemodels.uchi-
cago.edu/modtran/.
As the MODTRAN web application only models the longwave radiation, in a case
where the shortwave (SW) radiation was needed in this study, this was estimated by an-
other model, RRTM (standing for Earth's Energy Budget), again available as a web appli-
cation by the University of Chicago [
49
]. This simulates both LW and SW radiation fluxes,
both upward and downward.
3. Data
To compare MODTRAN results to observed radiation profiles in the atmosphere, we
need radiosonde data. While radiosondes are routinely made in several hundreds of sites
across the world, they typically measure temperature, humidity, pressure and wind. Ra-
diation radiosonde measurements are rare, yet it is useful to make at least a single com-
parison to get a general idea. Here we use a couple of radiosondes from a day and a night
flight on 23 September 2011, in cloud-free conditions, at the aerological station in Payerne,
Switzerland (6.9440° E, 46.8130°, +491 m a.s.l.). In these, in-situ measurements of down-
ward and upward radiation fluxes were taken through the troposphere and into the strat-
osphere, exceeding 32 km of altitude. They were presented by Philipona et al. [
50
] in
graphical form in their figure 2, which was digitized here to recover the measurements.
For the downwelling LW radiation flux, there have been numerous measurements at
specific sites and for a period of more than a century [18]. These were the basis for the
derivation of the empirical or semiempirical formulae for the calculation of the down-
welling radiation flux, discussed in Section 2.2. Considering that these formulae reflect
the data that was based upon, here we use the formulae, instead of the data, for compari-
sons with MODTRAN.
Information for the radiation fluxes at the top of the atmosphere (TOA), including
the LW fluxes, are provided by satellite instruments. These are available only for the 21st
century from the ongoing project Clouds and the Earth’s Radiant Energy System (CERES),
a part of NASA’s Earth Observing System, designed to measure both solar-reflected and
MODTRAN
[CO₂] = 800 ppm
[CO₂] = 400 ppm
Stefan-Boltzmann law
T = 300 K
T = 280 K
T = 260 K
T = 240 K
T = 220 K
0.5
0.4
0.3
0.2
0.1
0
0 500 1000 1500 2000
Wavenumber (cm–1)
Outgoing LW intensity (W m–2cm)
9
Earth-emitted radiation from the TOA (in CERES defined at the altitude of 20 km) to the
surface.
The specific product used here is the CERES SSF1deg monthly averaged TOA LW
radiative fluxes at a 1°-regional grid, constant-meteorology-temporally-interpolated [
51
-
53
]. The TOA fluxes are provided for clear-sky and all-sky conditions. Both these obser-
vational data are available online [
54
] and were retrieved here for their entire time span
of complete years, i.e., from January 2001 to December 2022 from the Terra platform. The
same CERES product provides information on clouds, which was also retrieved (monthly
averaged for both day and night).
In addition, the CERES project provides LW fluxes for the surface, both downwelling
and upgoing, through the product CERES_EBAF_Ed4.2, where EBAF stands for “Energy
Balanced and Filled” [
55
,
56
]. However these are computed gridded values rather than
observational data. They are publicly available [
57
] and were also retrieved here.
The CERES data are associated with considerable uncertainties. According to CERES
[
58
], the combined regional all-sky LW flux uncertainty is 2.4 W/m² and the daily regional
all-sky LW diurnal uncertainty is 8 W/m². According to CERES [
59
; Table 6.1], the uncer-
tainties in the 1°×1° regional monthly TOA fluxes are 4.6 W/m² for clear sky and 2.5 W/m²
for all sky. In addition, as also noted in CERES [59], with the most recent CERES instru-
ment calibration improvements, there still is a net imbalance of ~4.3 W/m², much larger
than the expected observed ocean heating rate which CERES assumes to be ~0.71 W/m².
It is noted though that, according to the calculations by Koutsoyiannis [23], the latter im-
balance value, again inferred from ocean heating data, is even lower, 0.37 W/m². The EBAF
dataset adjusts the observations to remove the above inconsistency. All this information
suggests that the observational uncertainties are far too high to allow calculations of
Earth’s imbalance and of temporal climatic changes, yet they are quite useful for the scope
of this study.
Additional atmospheric variables used here, namely temperature and water vapour
pressure, are taken from the ERA5 and NCEP/NCAR Reanalyses at monthly scale. ERA5
stands for the fifth-generation atmospheric reanalysis of the European Centre for Me-
dium-Range Weather Forecasts (ECMWF; ECMWF ReAnalysis). Its data are publicly
available for the period 1940 onwards at a spatial resolution of 0.5°. NCEP/NCAR stands
for Reanalysis 1 by the National Centers for Environmental Prediction (NCEP) and the
National Center for Atmospheric Research (NCAR). Its data are publicly available from
1948 to the present at a horizontal resolution of 1.88° ( 210 km). Both data sets can be
retrieved from the Climexp platform [
60
] and from the Physical Sciences Laboratory plat-
form of the US National Oceanic and Atmospheric Administration (NOAA) [
61
]. Finally
[CO₂] data were retrieved for the most well-known station, Mauna Loa, again from the
Climexp platform.
The gridded CERES SSF1deg TOA LW data are presented in Figure 2, averaged over
the period of observations. The cloud information (cloud area fraction) of the same data
product is shown in Figure 3. The zonal distribution of the LW radiation data are shown
in Figure 4, along with that of temperature, and, as expected, the distributions of the two
variables are similar. Figure 5, constructed from the data of Figure 4 shows that the torrid
zone (between 23.4° S and 23.4° N) contributes 43% of Earth’s outgoing LW radiation flux,
and together with the two temperate zones (between 23.4° and 66.6° S and N) radiates
94% of the total, leaving only 6% for the frigid zones (between 66.6° and 90° S and N).
Figure 6 better depicts the relationship of outgoing LW radiation flux and surface temper-
ature, for each month of the 22 years of data availability separately, but areally averaged
over geographical zones of 15° latitude. This graph allows two important observations.
First, the individual monthly values align very well with the average zonal distribution.
Second, at the high end of temperature variation, around 300 K, there appears a stagnancy,
or even regression of radiation with respect to temperature.
10
Figure 2 Geographical distribution of outgoing LW radiation averaged over the period of 2000 –
2022 as given by the CERES data: (upper) all sky; (lower) clear sky. Data retrieved from https://ceres-
tool.larc.nasa.gov/ord-tool/jsp/SSF1degEd41Selection.jsp; graph generated by
https://climexp.knmi.nl.
Figure 3 Geographical distribution of cloud area fraction averaged over the period of 2000 – 2022 as
given by the CERES data. Data source and graph generation as in Figure 2.
W/m2
%
11
Figure 4 Zonal distribution of LW radiation averaged over the period of 2000 – 2022 as given by the
CERES data: (upper) downwelling; (lower) outgoing; , the temperature zonal distribution, as given
by ERA5 Reanalysis is also plotted. Radiation data retrieved from https://ceres-
tool.larc.nasa.gov/ord-tool/jsp/SSF1degEd41Selection.jsp for outgoing and https://ceres-
tool.larc.nasa.gov/ord-tool/jsp/EBAF42Selection.jsp for downwelling; temperature data retrieved
from, and graph generated by, https://climexp.knmi.nl.
Figure 5 Graphical depictions of the relative intensity of outgoing LW radiation averaged over the
period of 2000 – 2022 as given by the CERES data, as a function of the latitude; (left) ratio of intensity
at latitude φ (S or N) to that at the equator; (right) cumulative flux between latitudes φ S and φ N
to the total outgoing radiation.
220
230
240
250
260
270
280
290
300
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
420
-90 -60 -30 0 30 60 90
Temperature (K)
Clear sky LW downwelling flux (W/m2)
Latitude, φ(°)
220
230
240
250
260
270
280
290
300
310
120
140
160
180
200
220
240
260
280
300
-90 -60 -30 0 30 60 90
Temperature (K)
Clear sky LW outgoing flux (W/m2)
Latitude, φ(°)
LW flux, clear sky
LW flux, all sky
Temperature
0
0.2
0.4
0.6
0.8
1
015 30 45 60 75 90
LW flux flux per ° of latitude relative to equator
Latitude, φ(°)
LW flux, clear sky
LW flux, all sky
Tropic circles, 23.4° (43%)
Polar circles, 66.6° (94%)
0
0.2
0.4
0.6
0.8
1
015 30 45 60 75 90
Fraction of total LW flux from -φto φ
Latitude, φ(°)
LW flux, clear sky
LW flux, all sky
12
Figure 6 Depiction of monthly variability of outgoing LW radiation, as given by the CERES data;
each point is the monthly value, spatially averaged over a geographical zone of 15° latitude, during
the period of 2000 – 2022; the zonal distributions (temporal averages for the entire period as in Fig-
ure 4) are also shown for each hemisphere (SH and NH).
4. Macroscopic Relationships
4.1 Clear-sky Relationships
Faithful to Dooge’s [
62
] spirit of “looking for hydrologic laws”, i.e., simple macro-
scopic laws for phenomena whose details are complex, this section tries to establish rela-
tionships between the downwelling and outgoing LW radiation fluxes with the variables
that influence it, namely, temperature, water vapour pressure, carbon dioxide concentra-
tion and cloudiness. To this aim it primarily uses MODTRAN outputs and also cloudiness
information.
A series of systematic MODTRAN runs around the values of the standard tropical
profile, i.e., [CO₂] = 400 ppm and for surface T = 299.7 K and = 19 hPa, with ranges of
200 – 800 ppm for [CO₂], ± 2 K for T and ± 10% for , gave the results shown in contour
graphs in Figure 7 for the downwelling LW flux and in Figure 8 for the outgoing flux.
Both figures suggest that the relationships among the involved variables are simple and
that macroscopic representations are possible. Figure 7 shows a linear effect of and a
logarithmic one of [CO₂] on the variation the downwelling LW flux. In addition, it shows
that a ± 10% change in is larger than a doubling or halving of [CO₂]. Similar are the
results of Figure 8, but with smaller differences of the effects of the two factors.
120
140
160
180
200
220
240
260
280
300
320
220 230 240 250 260 270 280 290 300 310
Outgoing LW flux (W/m2)
Temperature (K)
90°S-75°S 75°N-90°N
75°S-60°S 60°N-75°N
60°S-45°S 45°N-60°N
45°S-30°S 30°N-45°N
30°S-15°S 15°N-30°N
15°S-0° 0°-15°N
Zonal, SH Zonal, NH
13
Figure 7 Changes in the downwelling LW radiation, , as calculated by MODTRAN, due to
changes in temperature (converted to blackbody radiation, ), and (upper) water vapour pres-
sure () and (lower) carbon dioxide concentration ([CO₂]). The calculations were made for the trop-
ical profile, no clouds and default other settings. For the upper graph it was assumed [CO₂] = 400
ppm and for the lower graph = 19 hPa. Notice that the scale of the vertical axis is linear in the
upper graph (with range ±10% of the central value) and logarithmic in the lower graph (with range
from half to twice the central value).
Figure 8 As in Figure 7 but for the outgoing LW radiation, .
445 448 451 454 457 460 464 467 470
[CO₂] (ppm)
σT4 (W/m²)
LO
800
400
200
ea(hPa)
380-385 375-380 370-375 365-370 360-365 355-360 350-355
LD(W/m²):
20.9
19.0
17.1
445 448 451 454 457 460 464 467 470
[CO₂] (ppm)
σT4 (W/m²)
285-290 290-295 295-300 300-305 305-310 310-315
LO
800
400
200
ea(hPa)
285-290 290-295 295-300 300-305 305-310
L
O
(W/m²):
20.9
19.0
17.1
14
Based on these preliminary results, several mathematical expressions were formu-
lated and fitted. The following equation was found to be the best for both the downwelling
and outgoing flux, and , respectively:
(21)
with = 400 ppm. This includes two groups of parameters to be optimized, namely
(a) dimensional, , , and
with units [L], [T], and [], respectively, and (b) dimen-
sionless , , and . Excepting the last one, referring to the cloud area fraction, C,
and discussed in Section 4.2, all others were optimized based on clear-sky MODTRAN
results, and their values are contained in Table 1.
Table 1 Fitted parameters of Equation (21) for ranges of temperature 247.2 – 309.7 K (–26.0 – 36.6
°C), water vapour pressure 1.08 – 20.9 hPa and CO₂ concentration 200 – 800 ppm.
Sign†
(W/m²)
(K)
(hPa)
(ppm)‡
§
Downwelling,
+
27
181
6.36
400
4.5
1
0.015
0.34
Outgoing,
–
1
55
0.00302
400
3.5
0.5
0.015
0.15
† Specific sign to replace ± in Equation (21).
§ These are global values; for values of outgoing radiation per geographical zone see Figure 13.
‡ Not necessary to optimize.
As seen in Table 2, the performance measures of the fitting are very good and hence
the equation is a good macroscopic representation of the MODTRAN results. The good
performance is also seen graphically in Figure 9 for downwelling flux and Figure 10 for
outgoing flux.
Table 2 Performance indices of Equations (21) (with parameters as in Table 1) and (22), fitted for
ranges of temperature 247.2 – 309.7 K (–26.0 – 36.6 °C), water vapour pressure 1.08 – 20.9 hPa and
CO₂ concentration 200 – 800 ppm, and clear sky.
Equation, variable
Range of
(W/m2)
RMSE† in
(W/m2)
Explained
variance§ of
(%)
RMSE† in
relative error
(%)
Maximum absolute
relative error
(%)
(21), downwelling,
140.0 – 419.5
1.3
99.97
0.5
1.9
(21), outgoing,
170.6 – 351.1
1.3
99.79
0.7
1.9
(22), outgoing, from
170.6 – 351.1
3.6
99.24
1.5
4.5
† Root mean square error.
§ Also known as Nash-Sutcliffe performance index.
15
Figure 9 Comparison of downwelling LW radiation flux, , as computed by MODTRAN and by
Equation (21).
For completeness, an additional equation was formulated, which directly relates the
outgoing to the downwelling radiation. It is very simple:
(22)
where to calculate the surface radiation flux the emissivity taken . The perfor-
mance of Equation (22) is also good, albeit inferior to that of Equation (21), as seen in Table
2 and Figure 11. This equation shows that the outgoing and downwelling flux are closely
(and negatively) related to each other.
100
150
200
250
300
350
400
450
100 150 200 250 300 350 400 450
Downwelling LW flux from equation (W/m²)
Downwelling LW flux from MODTRAN (W/m²)
200 ppm
400 ppm
800 ppm
Equality line
16
Figure 10 Comparison of outgoing LW radiation flux, , as computed by MODTRAN and by Equa-
tion (21).
Figure 11 Comparison of outgoing LW radiation flux, , as computed by MODTRAN and by Equa-
tion (22), based on the downwelling flux, .
100
150
200
250
300
350
400
450
100 150 200 250 300 350 400 450
Outgoing LW flux from equation (W/m²)
Outgoing LW flux from MODTRAN (W/m²)
200 ppm
400 ppm
800 ppm
Equality line
100
150
200
250
300
350
400
450
100 150 200 250 300 350 400 450
Outgoing LW flux from equation (W/m²)
Outgoing LW flux from MODTRAN (W/m²)
200 ppm
400 ppm
800 ppm
Equality line
17
4.2 Effect of Clouds
MODTRAN offers the possibility to model different types of clouds, in addition to
clear-sky conditions. Figure 12 shows an example for cumulus clouds and the tropical
profile. In particular, it shows that Equation (21) with 0.904 (standing for the
ratio of outgoing radiation under cloudy sky to that of clear sky) also represents well the
case of cumulus clouds without changing the parameters. Similar performance appears
for other types of clouds, but with different ratios, which are shown in Table 3 for the
different types of clouds and locality profiles.
Figure 12 Comparison of outgoing LW radiation flux, , as computed from MODTRAN and from
Equation (21), fitted only for the tropical profile and for T = 299.7 ± 2 K, = 19 ± 1.9 hPa, and [CO₂]
range of 200 – 800 ppm. The parameters of Equation (21) were reoptimized for these ranges, and the
values = 56.2 K and
0.00683 hPa were found, while all others remained the same as in Table
1. The case of cumulus clouds is also plotted in the graph, with abscissae as derived by MODTRAN
and ordinates equal to the values of Equation (21) for clear sky multiplied by 0.904.
Table 3 Ratios of outgoing LW radiation under cloudy sky to that for clear sky, for the indicated
types of clouds and locality profiles, as determined by MODTRAN for default settings (the lowest
value is highlighted in bold).
Locality profile →
Cloud type ↓
Tropical
Midlatitude
summer
Subarctic
summer
Midlatitude
winter
Subarctic
winter
Cumulus
0.904
0.899
0.889
0.910
0.95
Altostratus
0.901
0.896
0.887
0.907
0.948
Stratus
0.966
0.973
0.962
0.973
1.016
Stratus/Stratocumulus
0.939
0.945
0.931
0.950
0.989
Nimbostratus
0.98
0.984
0.977
0.984
1.024
Standard Cirrus
0.93
0.939
0.957
0.95
0.974
NOAA Cirrus
0.937
0.945
0.963
0.956
0.98
However, it is important to test whether these ratios agree with observed ones. From
the CERES data sets, the global average LW radiation flux for the entire observation
250
260
270
280
290
300
310
320
250 260 270 280 290 300 310 320
Outgoing LW flux, LO, from equation (W/m²)
Outgoing LW flux, LO, from MODTRAN (W/m²)
Clear sky
Cumulus clouds
Equality line
18
period, 2001 – 2022, is
268.2 W/m² for clear sky and
239.5 W/m² for all sky, and
give a ratio of 0.892. On the other hand, the global average cloud area fraction, again cal-
culated from the CERES data, is 0.671. We assume that the following approximation
holds:
(23)
from which we find (1 – 0.892) / 0.671 = 0.161. For , we would have
0.839. Comparing the latter value to those in Table 3, we understand that the MODTRAN
model severely underestimates the effect of clouds, as even the least value of the ratio in
the Table (0.887 for altostratus clouds and the subarctic summer profile) is too high in
view of an average of 0.839. Therefore, to model the effect of clouds we use the CERES
data rather than the MODTRAN results, based on the simple relationships:
(24)
where and
denote the outgoing LW radiation flux for clear sky and cloudiness C,
respectively (and likewise for the downwelling, and
) and parameter, which may
take different values for different situations. These relationships are already incorporated
in Equation (21).
To determine the parameter for the outgoing LW radiation we investigate the
CERES data (clear and all sky, and cloud fraction, globally and in the five different geo-
graphical zones, as shown in Figure 13. The resulting values are also shown in the Fig-
ure and vary from 0.18 in the torrid zone to 0.09 in the frigid zones. Notable is the large
spread of the observations, with a variation range of 0 to about twice the estimated mean
slope (i.e.,
from 0 to 0.30).
The data used in this study do not allow direct estimation of for downwelling flux
and thus we use values from literature. Dingman [
63
, p. 189] suggested a value of 0.4,
Jacobs [
64
, p. 108] estimated a value of 0.33 and Lhomme et al. [
65
] a value of 0.34. (This
could be also taken as 0.37 as the formula in their equation (13) can be written as is
.) Of these values, here
we use as this value was based on the most extensive data set and is most recent.
Other researchers give different formulae [
66
(p.142),41,43], but here we preferred the sim-
plest linear formulation.
19
Figure 13 Effect of clouds to outgoing radiation, expressed as
, i.e. the deviation from 1
of the ratio of longwave radiation for all sky (
) to that for clear sky (
), vs. the cloud fraction,
C. The slopes of homogenous linear regressions, are also shown. Each plotted point corresponds
to one of the 64 800 grid points of the CERES grids of clear-sky and all-sky radiation (Figure 2), and
cloud area fraction (Figure 3), where the average over the period 2001-2022 was taken.
5. Test Results
5.1 Radiation Flux Profiles
As already mentioned (Section 3), the most appropriate way to test the validity of a
model that determines the LW radiation, such as MODTRAN, would be to compare its
results to observed radiation profiles. As described in Section 3, here we make a single
comparison to get a general idea, using the two radiosondes launched on 23 September
2011 at Payerne, Switzerland, and reported in the study by Philipona et al. [50], whose
LW radiation profiles were digitized here. As reported in the study, the surface-emitted
LW upward radiation was about 445 W/m² during the day, with a remarkably strong
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
1 - LWAS/LWCS
Cloud fraction, C
Torrid
aC= 0.18
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Cloud fraction, C
Global
aC= 0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
1 - LWAS/LWCS
Cloud fraction
Temperate, NH
aC= 0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Cloud fraction
Temperate, SH
aC= 0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
1 - LWAS/LWCS
Cloud fraction, C
Frigid, NH
aC= 0.09
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Cloud fraction, C
Frigid, SH
aC= 0.09
20
decrease during the first 1 km, and 380 W/m² during the night. A surface temperature of
11 °C (284.2 K) is reported in the study for 23 September 2011, without clarifying whether
this corresponds to the day or night radiosonde. Using MODTRAN with the standard
midlatitude summer profile, we find that at the altitude of +0.5 km, the values of 445 W/m²
and 380 W/m² are achieved for a temperature offset of +5.5 K (297.5 K at +0.5 km) and –6.2
K (285.8 K at +0.5 km, close to the reported 284.2 K), respectively. However, the former
temperature offset gives too high temperature at higher altitudes, inconsistent with the
observed strong decrease across the first 1 km. Therefore for the comparison of the day
profile with MODTRAN we did not use any temperature offset, while for the night case
we assumed an offset of –6.8 K (285.1 K, midway of the values 284.2 K and 285.8 K).
Figure 14 depicts the radiosonde profiles, compared with the MODTRAN results ob-
tained with the above assumptions. For the night radiosonde, there is a good agreement,
yet the differences between observations and MODTRAN results are ±25% for the down-
ward flux and ±4% for the upward flux. For the day radiosonde, the differences are sub-
stantial, particularly above 7 km. Philipona et al. [50] attribute these differences to the
thermal longwave radiation from the Sun—a plausible interpretation. Excepting the latter
factor, we may deem that MODTRAN represents the relevant processes satisfactorily.
Figure 14 Comparison of LW radiation components calculated by MODTRAN to radiosonde meas-
urements by Philipona et al. [50].
5.2 Downwelling Radiation
As already mentioned, we use the empirical or semi-empirical formulae of down-
welling radiation to test the MODTRAN results. These formulae are expressed in terms of
emissivity
(Section 2.2). Figure 15 shows that the emissivity calculated by the
0
5
10
15
20
25
30
35
0.01 0.1 1
Altitude (km)
Ratio of downward to upward LW radiation flux
0
5
10
15
20
25
30
35
50 100 150 200 250 300
Altitude (km)
Net upward LW radiation flux (W/m²)
0
5
10
15
20
25
30
35
110 100
Altitude (km)
Downward LW radiation flux (W/m²)
0
5
10
15
20
25
30
35
250 300 350 400 450
Altitude (km)
Upward LW radiation flux (W/m²)
Day, radiosonde
Night, radiosonde
Day, MODTRAN
Night, MODTRAN
21
macroscopic relationship of Equation (21) perfectly agrees with that directly calculated by
MODTRAN (explained variance, EV = 99.9%). As seen in Figure 16, the agreement is not
so good if we compare MODTRAN’s emissivity with those of the formulae in Section 2.2.
The best agreement (EV = 61.6%) is seen for Prata’s Equation (15) followed by Brutsaert’s
Equation (13) (EV = 39.5%) and the FAO Penman-Montieth’s Equation (20) (EV = 1%).
The Brunt / Penman Equation (18) gives an EV < 0, meaning that a single value equal to
the mean performs better than the relationship in representing the compared data.
Note that if the group of points corresponding to subarctic winter with temperatures
lower than 270 K is excluded, Brutsaert’s equation has the best performance of all, with
an explained variance of 75%. Overall, if we assume that the empirical or semi-empirical
formulae are closer to reality than MODTRAN because they are based on data and if we
exclude the subarctic winter data (and the Brunt / Penman equation) we conclude that
MODTRAN underestimates the downwelling radiation flux.
Another comparison is made in Figure 17 for the downwelling LW radiation flux vs.
temperature, as calculated by MODTRAN and the CERES EBAF zonal distribution shown
in Figure 4. For the former all five locality profiles are used with default settings as well
as with temperature offsets from the default values of up to ±25 K. We recall that CERES
EBAF data are not actually measurements but computed results. Yet the graph suggests a
tendency of MODTRAN to underestimate the downwelling radiation, a finding similar to
that in the comparison with the emissivity formulae.
Figure 15 Comparison of emissivity calculated by MODTRAN to that calculated by Equation (21).
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Emissivity εa, from vapour pressure
Emissivity εa, from MODTRAN
EV = 99.9%
Proposed
22
Figure 16 Comparison of emissivity calculated by MODTRAN to that calculated by each of the in-
dicated formulae; the explained variance (EV) is also shown for each case.
Figure 17 Downwelling LW radiation flux vs. temperature, as calculated by MODTRAN for the five
indicated locality profiles and default settings (points with crosses) as well as with temperature
offsets from the default values of up to ±25 K (all other points), in comparison to the CERES EBAF
zonal distribution shown in Figure 4.
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Emissivity εa, from vapour pressure
Emissivity εa, from MODTRAN
EV = 61.6%
Prata
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Emissivity εa, from vapour pressure
Emissivity εa, from MODTRAN
EV = 39.5%
Brutsaert
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Emissivity εa, from vapour pressure
Emissivity εa, from MODTRAN
EV = 1%
FAO Penman-Montieth
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Emissivity εa, from vapour pressure
Emissivity εa, from MODTRAN
EV = -77.3%
Brunt / Penman
80
120
160
200
240
280
320
360
400
440
480
520
220 240 260 280 300 320
Downwelling LW flux (W/m2)
Temperature (K)
CERES EBAF, Northern Hemisphere
CERES EBAF, Southern Hemispher e
MODTRAN, Tropical
MODTRAN, Midlatitude summer
MODTRAN, Midlatitude winter
MODTRAN, Subarctic summer
MODTRAN, Subarctic winter
23
5.3 Outgoing Radiation
For the outgoing radiation, a comparison similar to that in Figure 17 is made. It is
presented in Figure 18 and is now stronger and more meaningful, as the CERES outgoing
LW flux is observed, rather than computed. The graph shows that at the default temper-
ature values of each of the MODTRAN profiles there is almost perfect agreement between
CERES data and MODRTAN results, and this is also extended for negative temperature
offsets in MODRTAN. However for positive temperature offsets, MODRTAN overesti-
mates the flux.
Figure 18 Outgoing LW radiation flux vs. temperature, as calculated by MODTRAN for the five
indicated locality profiles and default settings (points with crosses) as well as with temperature
departing from the default values by up to ±25 K (all other points), in comparison to the observed
zonal distribution (CERES SSF1) as shown in Figure 4 and Figure 6.
As another test, we use time series of meteorological variables and [CO₂] for the 22
years of CERES data availability averaged over the torrid zone, which are shown in Figure
19. For temperature and water vapour pressure, both ERA5 and NCEP/NCAR reanalyses
data are used, which, as shown in the graphs have differences from each other, particu-
larly in the latter variable. Based on these time series and Equation (21), the model results,
which correspond to MODTRAN, are calculated and plotted in Figure 20. In the upper
panel of the Figure, substantial differences are seen in the mean levels of the clear-sky LW
outgoing radiation, both between the time series of the two reanalyses and the latter with
the CERES data. These can be attributed mostly to uncertainty in Reanalyses results and
their representation of reality, as indicated in the differences between the two. If we ex-
clude the effect of the mean level differences, the cross-correlation coefficients with the
CERES data are reasonable, i.e. 0.67 for ERA5 and 0.69 NCEP/NCAR.
In the lower panel of Figure 20, we use the CERES clear-sky time series, as shown in
the upper panel, and multiply it by the quantity of Equation (21), with
(the value of the torrid zone) and cloudiness C as in the time series of the lowest panel
of Figure 19. The attained results compare well with the all-sky CERES time series in terms
of mean level, even though the cross-correlation between the two plotted time series (–
0.15) does not suggest correspondence of the individual monthly values.
120
160
200
240
280
320
360
220 240 260 280 300 320
Outgoing LW flux (W/m2)
Temperature (K)
Data, Northern Hemisphere
Data, Southern Hemisphere
Model, Tropical
Model, Midlatitude summer
Model, Midlatitude winter
Model, Subarctic summer
Model, Subarctic winter
24
Figure 19 Time series of the indicated variables, averaged over the torrid zone. The values plotted
are monthly averages, while the 11-year averages are also plotted in thicker lines.
297
298
299
300
2000 2005 2010 2015 2020 2025
Temperature (K)
ERA5 Reanalysis
NCEP/NCAR Reanalysis
22
23
24
25
26
27
28
29
2000 2005 2010 2015 2020 2025
Water vapour pressure (hPa)
ERA5 Reanalysis
NCEP/NCAR Reanalysis
360
370
380
390
400
410
420
430
2000 2005 2010 2015 2020 2025
[CO₂] Mauna Loa (ppm)
0.58
0.59
0.6
0.61
0.62
0.63
0.64
0.65
0.66
0.67
2000 2005 2010 2015 2020 2025
Total cloud fraction -CERES (W/m2)
25
Figure 20 (upper) Time series of outgoing clear-sky LW radiation flux, averaged over the torrid
zone, as given by CERES data and as predicted by Equation (21) with temperature, water vapour
and [CO₂] data from the ERA5 and NCEP/NCAR Reanalyses. (lower) Time series outgoing all-sky
LW radiation flux, averaged over the torrid zone, as given by CERES data and as predicted by the
CERES clear-sky data and Equation (21) with CERES cloud area fraction. The values plotted are
monthly averages, while the 11-year averages are also plotted in thicker lines.
Both in Figure 19 and Figure 20, the average values of the time series are shown for
two 11-year subperiods, 2001 – 2011 and 2012 – 2022. It is seen that there were increases
in the second period in all variables (very large in CO2 and very slight in all-sky outgoing
radiation) except in the clear-sky radiation, where a slight decrease (–0.08 W/m2) is seen.
It is interesting to investigate if the latter decrease is explained by MODTRAN. The results
of the related calculations are shown in Table 4. Using any of its default profiles, MOD-
TRAN results in increase of outgoing clear-sky LW radiation (from +0.38 to +0.60 W/m2),
opposite to what is seen in the CERES data.
Table 4 Comparison of observed data and MODTRAN results for the average conditions of each of
the indicated two periods. The observed data are global averages. The MODTRAN results are for
the average observed temperature and [CO₂] of each period, and for the same water vapour scales
281
282
283
284
285
286
287
288
289
290
291
292
293
294
2000 2005 2010 2015 2020 2025
Outgoing clear sky LW flux (W/m2)
CERES data
Model, ERA5 data
Model, NCEP/NCAR data
252
254
256
258
260
262
264
2000 2005 2010 2015 2020 2025
Outgoing all sky LW flux (W/m2)
CERES data
Model with clear sky CERES data
26
in the two periods, estimated so as to (approximately) match the outgoing clear-sky LW flux of
CERES for the first period, while holding fixed relative humidity for the two periods.
Period →
2001-11
2012-22
Difference
Observations (averages over each period)
Temperature from ERA5 (K)
287.21
287.49
+0.28
[CO₂] from Mauna Loa (ppm)
381.83
406.48
+24.65
Outgoing TOA (20 km) clear-sky LW from CERES (W/m2)
268.33
268.27
–0.08
Outgoing clear-sky LW radiation by MODTRAN at 20 km altitude (W/m2)
Tropical profile, water vapour scale 0.82
268.34
268.72
+0.38
Midlatitude summer profile, water vapour scale 1.08
268.34
268.78
+0.44
Midlatitude winter profile, water vapour scale 1.14
268.28
268.78
+0.50
Subarctic summer profile, water vapour scale 2.56
268.31
268.91
+0.60
The above results from CERES data, which are for the torrid zone only, are similar to
those for global averages, presented by Koutsoyiannis and Vournas [18; Appendix B]. The
latter study also examined SW radiation data and found a decrease of total outgoing ra-
diation, which is consistent with the increased atmospheric temperature. This decrease of
outgoing radiation can hardly be attributed to increased [CO₂] but it can be related to
water vapour and cloud profiles. The effect of CO₂ is trumped by the effect of clouds,
which is consistent with the major role of water on climate and the minor one of CO₂.
5.4 Final Assessment
The above tests illustrate the high uncertainties not only in the CERES LW radiation
data, but also in the other atmospheric variables, and the relationships among them and
the LW radiation, as represented in MODTRAN. The uncertainties do not allow accurate
representation of quantities calculated as differences between different variables or be-
tween the same variables in different periods, such as in attributing changes. On the other
hand, the macroscopic behaviour of MODTRAN seems consistent with what is observed
for clear sky, and therefore MODTRAN is suitable for the scope of this paper, which is the
investigation of the relative importance of carbon dioxide and water in the greenhouse
effect, as detailed in the next section. As regards clouds, MODTRAN seems to underesti-
mate their effect, but by using the cumulus or altostratus cloud conditions, we get results
close to reality for average all-sky conditions.
6. Comparison of H₂O and CO₂
6.1 Imaginary-world Conditions
Investigating imaginary-world conditions seems pointless, yet we include it for the
reasons explained in the Introduction—the fact that several popular narratives are based
on imaginary-world conditions. The imaginary-world conditions we examine are the
most extreme ones, starting from the case that [CO₂] is totally absent in the atmosphere
and ending to the case where the atmosphere is composed of merely CO₂. Extreme ranges
are also examined for other greenhouse gases, including water vapour.
Figure 21 compares detailed MODTRAN outputs for the default case of the tropical
profile, with [CO₂] = 400 ppm (and T = 299.7 K, = 298.49 W/m², water vapour scale = 1;
red curves), compared to two cases in which CO₂ is totally absent (blue curves). In the left
panel the water vapour is kept at the same level as in the default case. In order for the
outgoing radiation to retain its default value ( = 298.49 W/m²), the temperature must be
lowered by 15 K (T = 284.7 K). Notably, by also zeroing water vapour, to match the total
outgoing LW flux of the red curve ( = 298.49 W/m²), the temperature should be lowered
by 28.5 K (T = 271.2 K, case not shown in the figure but easily imagined in the middle
between the smooth curves of 260 and 280 K). The right panel shows that the temperature
can remain at its default value (T = 299.7 K) if water vapour scale is increased to 1.3, ac-
companied by cumulus clouds. Note that the 30% increase of the water vapour scale does
not violate thermodynamic laws, as the vapour pressure remains below the
27
thermodynamic limit. Therefore, assertions that the terrestrial greenhouse effect would
collapse without CO₂ (see Introduction) are false.
Figure 21 Output of the MODTRAN model, with the red curves in both panels produced for default
settings (for the standard tropical atmospheric profile; T = 299.7 K; [CO₂] = 400 ppm; = 298.49
W/m²), and the blue curves produced for zero concentrations of all greenhouse gases except water
vapour, assuming constant relative humidity and conditions such as to much the total outgoing LW
flux of the red curves, namely: (left) temperature lower than default by 15 K (T = 284.7 K); (right)
temperature equal to the default (T = 299.7 K) but water vapour scale increased to 1.3, accompanied
by cumulus clouds.
The fact that the greenhouse effect would not collapse without CO₂ is also shown in
Figure 22 (left panel), which gives a more macroscopic picture for both downwelling and
outgoing LW flux, under constant temperature equal to that of the standard tropical at-
mospheric profile (T = 299.7 K) and zero concentrations of all greenhouse gases except
water vapour. It is seen than even without any change in the water vapour profile, the
difference of LW radiation flux from the value at the default settings of [CO₂] = 400 ppm
is only –3%. With a 10% increase of the water vapour and zero [CO₂], the downwelling
radiation matches that for [CO₂] = 400 ppm.
Additional imaginary-world conditions are shown in Table 5. It can be seen there
that by removing the entire quantity of atmospheric CO₂ we can achieve the same level of
greenhouse effect, as produced by [CO₂] = 400 ppm and zero water vapour, with merely
0.15% of the current H₂O level in terms of upward and downward LW heat flux at the
surface, or 4% in terms of the outgoing LW heat flux at 100 km altitude. On the contrary,
it is impossible to approach the values of greenhouse effect achieved by merely H₂O in
the atmosphere at the current level, by removing it and replacing it with CO₂. Even in an
atmosphere entirely composed by CO₂ (i.e. 1 000 000 ppm or 2500 times the current CO₂
concentration) we cannot approach the greenhouse values achieved by merely the current
level of atmospheric H₂O. On the other hand, we can easily achieve the greenhouse effect
level of an atmosphere entirely composed by CO₂ with an atmosphere free of CO₂ and
with only 20% of the current atmospheric H₂O for the downwelling radiation.
For completeness, we note that MODTRAN does not modify the temperature profile
in case that water vapour is removed. It assumes a temperature gradient of 6.5 K/km, as
in the standard atmosphere, or lower. However, without water vapour the temperature
gradient would be the dry adiabatic one, i.e. 9.8 K/km (see Figure 25 discussed in Section
7), which signifies another effect of water vapour on the atmospheric processes. And of
0 500 1000 1500 2000
Wavenumber (cm–1)
MODTRAN
[CO₂] = 0ppm
[CO₂] = 400 ppm
Stefan-Boltzmann law
T = 300 K
T = 280 K
T = 260 K
T = 240 K
T = 220 K
0.5
0.4
0.3
0.2
0.1
0
0 500 1000 1500 2000
Wavenumber (cm–1)
Outgoing LW intensity (W m–2cm)
28
course, without water vapour, clouds would not exist and the surface cooling effect would
be larger.
Figure 22 Difference of LW radiation flux from the values at the default settings of the standard
tropical atmospheric profile (T = 299.7 K; [CO₂] = 400 ppm; = 369.26 W/m²; = 298.49 W/m²), as
calculated by MODTRAN for the cases of (left) zero concentrations of all greenhouse gases except
water vapour, assuming constant relative humidity and (right) as left but with [CO₂] = 200 ppm.
Table 5 Results of MODTRAN calculations for temperature of 288 K (the value of current global
temperature used by Brutsaert [33] and tropical profile and extreme (imaginary-world) cases of
greenhouse gas concentrations.
[CO₂] relative to
the default
value of 400
ppm
Water vapour
scale relative to
the default
tropical profile
Other greenhouse
gases concentration
relative to default
Downward IR
heat flux at
surface (W/m²)
Upward IR heat
flux at surface
(W/m²)
Outgoing LW
radiation flux at 100
km altitude (W/m²)
1
1
1
325.6
381.5
249.5
1
1
0
324.0
381.5
256.7
0
0
0
1.7
380.3
379.0
1
0
0
68.2
380.3
340.7
2500*
0
0
215.7
381.5
257.1
0
0.0015
0
78.8
380.3
366.4
0
0.04
0
157.8
380.6
340.7
0
0.2
0
212.5
380.9
319.3
0
1
0
319.7
381.5
284.7
0
2.2
0
372.1
381.8
257.7
* In this case the atmosphere is composed merely by CO₂ (2500 × 400 = 1 000 000 ppm = 1).
6.2 Realistic Conditions
Putting aside the imaginary-world cases of an atmosphere without CO₂ or one
merely composed of CO₂, we may assume a realistic lowest value of [CO₂] of the order of
200 ppm. Redrawing the left panel of Figure 22 under this minimal value, instead of zero,
we get what appears in the right panel. It is obvious that, in comparison to the [CO₂] level
of 400 ppm the differences are small and can be easily counterbalanced by slight changes
in water vapour pressure.
Another view of the same is provided by Figure 23, where the difference from the
[CO₂] level of 400 ppm for two cases, doubling or halving this level, are presented. The
differences are of the order of 1% and do not reach 2%.
-15
-10
-5
0
5
10
15
0 5 10 15 20 25 30
Percent difference (%)
Percent increase of water vapour pressure
-15
-10
-5
0
5
10
15
0 5 10 15 20 25 30
Percent difference in LW flux (%)
Percent increase of water vapour pressure
Downwelling, clear sky
Outgoing, clear sky
Outgoing, cumulus clouds
29
Figure 23 Difference of LW radiation flux for the indicated conditions of [CO₂] from the respective
values at the default settings of the standard tropical atmospheric profile ( T = 299.7 K; [CO₂] = 400
ppm; = 369.26 W/m²; = 298.49 W/m²), as calculated by MODTRAN.
For a more systematic investigation, we may determine the relative importance of
each of the factors influencing the LW radiation flux, say factor F, by considering the rel-
ative change
produced by a relative change
in the factor and taking the ratio:
(25)
where
and
is the log-log partial derivative of L with re-
spect to F (defined in [29], p. 97).
Equation (21) allows analytical determination of the log-log derivatives and hence
the relative importance of each of the factors . This is made in Table 6.
MODTRAN also includes other greenhouse drivers with minor importance, i.e. CH4, trop-
ospheric ozone, stratospheric ozone, and freon, which were not modelled in the above
analyses. To calculate their bulk contribution, we increased each of the default values in
MODTRAN by 5% (
, where AO stands for “all other”), calculated
by
MODTRAN for the tropical and subarctic summer profile (for comparisons) and applied
Equation (25) to calculate
, which resulted in values also included in Table 6; the rela-
tively high value in the outgoing flux is primarily due to the influence of the stratospheric
ozone.
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
275 280 285 290 295 300 305 310 315 320 325
Percent difference in LW flux (%)
Temperature (K)
[CO₂] × 2, Outgoing
[CO₂] × 2, Downwelling
[CO₂] / 2, Outgoing
[CO₂] / 2, Downwelling
30
Table 6 Relative changes of LW radiation fluxes at standard conditions, equal to the global averages,
T = 288.6 K, = 15.2 hPa, = 400 ppm, C = 0.671, as calculated analytically and numerically by
Equation (21).
Case of relative
change
Relative change
expression
Numerical value of relative change
Downwelling flux
Outgoing flux
3.18
4.44
0.207
–0.136
0.015
–0.015
0.186
–0.112
(see text)
0.006
–0.023
It can be readily found using the values in Table 6, that the relative importance of
water vapour over [CO₂] is 0.207/0.015 = 13.8 times for the downwelling flux, and
times for the outgoing flux. The relative importance of clouds
over [CO₂] is 0.186/0.015 = 12.4 times for the downwelling flux, and (–0.112)/(–0.015) = 7.5
times for the outgoing flux. In other words, each of the related factors, water vapour and
clouds, is an order of magnitude more important than [CO₂] in terms of the greenhouse
effect.
Considering all above factors, the equation of total differential (with standard form
) can
readily be written as
(26)
or,
(27)
The first term in the righthand side,
, is by far the most important factor determining
the LW flux. The other terms describe the greenhouse effect. Excluding the first term, i.e.,
setting , Equation (27) allows breakdown of the relative importance of
the different factors. This is presented in Figure 24, where the compound importance of
water vapour and clouds is 95% and 87% for the downwelling and outgoing radiation,
respectively, while that of [CO₂] is 4% and 5%, respectively.
These results differ substantially from those of Schmidt et al. [
67
], who, using a dif-
ferent methodology, attributed 75% to water vapour and clouds and 19% to CO₂. Our
results are closer to an example given by Brooks [
68
], in which the contribution of the CO2
bands is about 1:8 compared to water vapour, without considering the clouds.
For completeness it must be noted that clouds also affect the planet’s albedo and the
incoming solar radiation, reducing it, but such an analysis is outside the scope of this pa-
per, which is focused on the greenhouse effect.
31
Figure 24 Contribution of the greenhouse drivers to the LW radiation fluxes, (left) downwelling
and (right) outgoing.
Finally, it is useful to estimate the changes that the increase of [CO₂] in a century,
from 300 to 420 ppm, may have caused. The results produced by Equation (21), as well as
those by direct run of MODTRAN, for this specific increase are shown in Table 7. The
former are deemed more reliable than the latter as they are based on a generalized equa-
tion representing all conditions, while the latter is based on a specific locality profile,
namely midlatitude summer profile (and altostratus clouds for the cloudy case). The
change in the downwelling radiation is estimated at 0.5% or lower, which could not be
discerned by observations, thus confirming the finding by Koutsoyiannis and Vournas
[18]. The change in the outgoing radiation is estimated also at 0.5% (but with negative
sign) or lower (compare also with Salby [
69
], p. 249), which also could not be discerned
by observations. Table 7 also shows the expected results for the case that [CO₂] increases
to 800 ppm. Now the change in the LW radiation flux is higher, 1.5%, and again could
hardly be detected by measurements in the future, in case that indeed [CO₂] reaches 800
ppm.
Table 7 Relative changes of LW radiation fluxes at standard conditions, equal to the global averages,
T = 288.6 K, = 15.2 hPa, C = 0.671 (or C = 0 for clear sky), and with the indicated values of [CO₂] as
calculated analytically and numerically by Equation (21). In parentheses are the values directly cal-
culated by MODTRAN assuming midlatitude summer profile matching the above values of T and
, and altostratus clouds for the cloudy case.
[CO₂] increase
Sky
Downwelling,
Outgoing,
mean*
% change
mean*
% change
From 300 to 420
Cloudy
1.93
(0.31)
382.4
(374.1)
0.5%
(0.1%)
–1.18
(–1.00)
235.1
(238.0)
–0.5%
(–0.4%)
From 300 to 420
Clear
1.57
(0.79)
311.3
(309.6)
0.4%
(0.3%)
–1.32
(–1.32)
261.5
(265.7)
–0.5%
(–0.5%)
From 300 to 800
Cloudy
5.63
(0.63)
384.2
(374.3)
1.5%
(0.2%)
–3.45
(–2.89)
234.0
(237.0)
–1.5%
(–1.2%)
From 300 to 800
Clear
4.59
(2.45)
312.8
(310.5)
1.2%
(0.8%)
–3.84
(–3.89)
260.2
(264.4)
–1.5%
(–1.5%)
* Geometric mean.
Our results in Table 7 are comparable to those of van Wijngaarden and Happer [
70
]
(corroborated in de Lange et al. [
71
]), who using a detailed representation for high-reso-
lution transmission molecular absorption database (HITRAN, a compilation of spectro-
scopic parameters that a variety of computer codes use to predict and simulate the trans-
mission and emission of light in the atmosphere) and satellite data, concluded that a
Water vapour
50%
Clouds
45%
[CO₂]
4%
All other
1%
Water vapour
48%
Clouds
39%
[CO₂]
5%
All other
8%
32
doubling of CO2 concentration (from 400 to 800 ppm) would result in a 3 W/m2 decrease
of radiation flux in the top of the atmosphere, which translates to –1.1%.
7. Discussion and Further Results
In view of the above results, we may revisit Lacis et al.’s [17] statements quoted in
the Introduction, leaving aside the fact they refer to imaginary-world conditions. Specifi-
cally, even if we removed the CO₂ from the atmosphere, again there would be new emis-
sions from volcanos and outgassing from the oceans, even if the biosphere was also re-
moved. In the relatively recent glacial periods, covered by Vostok proxy data, the CO₂
concentration did not fall below 180 ppm. This value is perhaps the absolute low for the
entire history of Earth. What would disappear from Earth in the imaginary-world case of
CO₂ removal is not the greenhouse effect but life as we know it. For plants may not survive
at CO₂ levels below 150 ppm [
72
] while without the photosynthesis performed by plants
the entire biosphere would collapse. On the opposite side, the increase in CO₂ is beneficial
for plant growth. As recently reported and as a result of the recent increase, global green-
ing is an “indisputable fact”, and even its rate has increased slightly [
73
].
Furthermore, even the 10% of the current atmospheric value of water vapour for
, given in the quoted statement by Lacis et al., would produce a greenhouse effect and
hence would imply the inequality , thus leading to absurd. That greenhouse effect
would not be 10% or close to it, but closer to its current magnitude. Indeed, according to
Brutsaert’s equation (13), for = 255/288 (with 288 K being the current average tem-
perature used by Brutsaert) and vapour pressure ratio = 0.1, the resulting emissivity
ratio would be (0.1 / (255/288))1/7 = 0.73. An emissivity means that we would
again have the greenhouse effect produced by water vapour. (See also Table 5 and its
discussion in Section 6.1.) And even in an “icebound Earth state”, thermodynamics implies
the presence of water vapour in the atmosphere, due to sublimation. Remarkably, though,
geological evidence presented by Veizer [
74
-
76
] suggests the presence of running water
as far back as we have a record, up to 3.8 or even 4.2 billion years, despite the much smaller
solar irradiance (the faint young sun puzzle). All these imply that the argument is mis-
taken and so is the popular result that is being widely reproduced.
The distinction between feedbacks and forcings, also appearing in the quoted state-
ments by Lacis et al. (and the bot), is problematic. Both H₂O and CO₂ are “native” and
abundant on Earth, and both are greenhouse gases, with the difference being that the for-
mer is the determinant, as already demonstrated. Calling CO₂ forcing and H₂O feedback
is like claiming that the tail wags the dog.
Water has more roles in climate than examined above. The greenhouse effect slows
down the rate of Earth’s cooling by LW radiation. However, Earth’s surface cooling and
its reaction, i.e., atmosphere’s warming, is not only due to LW radiation. According to
Trenberth et al. [
77
], the contribution of the LW radiation is 396 – 333 = 63 W/m2, a value
generally consistent with the results of this paper. This low value is due to the action of
greenhouse gases, and, as we have seen, it is dominated by the presence of water vapour
in the atmosphere. However, the greatest contribution in Earth’s surface cooling, namely
80 W/m2 [10, 77], is due to the latent heat from evaporation (phase change of water from
liquid to gaseous phase).
Surface cooling and atmosphere warming are reflected in the vertical profile of the
net upgoing radiation flux, seen in Figure 25. Here the SW radiation was also considered,
calculated by RRTM. The net radiation minus the absorbed SW radiation in the atmos-
phere is increasing with altitude. As, on the long term, the energy is not stored in the
atmosphere, the total heat transfer should be the same at all altitudes. The deficit of heat
transfer in low altitudes is recovered by sensible and latent heat transfer, which warm the
atmosphere. Figure 26 shows diagrammatically the contribution of each of these mecha-
nisms to Earth’s cooling and atmosphere’s warming.
33
Figure 25 (left) Vertical profile of net LW radiation flux for default conditions and standard tropical
atmospheric profile (T = 299.7 K; [CO₂] = 400 ppm; = 369.26 W/m²; = 298.49 W/m²), as calcu-
lated by MODTRAN, and its difference from the net SW profile, as calculated by the RRTM Earth's
Energy Budget (https://climatemodels.uchicago.edu/rrtm/). (right) MODTRAN’s standard tropical
atmospheric profile of temperature, compared with the dry adiabatic profile (with a gradient of 9.8
K/km); the arrows indicated the heating of the atmosphere due to the latent heat released by con-
densation of water vapour, accumulated over elevation.
Figure 26 Contribution of (left) the three mechanisms responsible for the cooling of Earth’s surface
and (right) the four mechanisms responsible for the warming of Earth’s atmosphere, based on the
energy balance by Trenberth et al. [77].
Interestingly, heat exchange by evaporation (and hence the latent heat transfer from
the Earth’s surface to the atmosphere) is the Earth’s natural locomotive, with the total
energy involved in the hydrological cycle being 1290 ZJ/year, corresponding to an energy
flux density of 80 W/m2. Compared to human energy production, the total energy of the
natural locomotive is 2100 times higher than that of the human locomotive [10].
In addition to regulating the LW radiation flux and the latent heat flux, water vapour
and clouds also regulate the SW radiation and Earth’s albedo. Other properties of water,
as listed in Koutsoyiannis [10], are also determinant for climate. First is its unique property
to appear on Earth in all three phases and in different formations, with spectacular differ-
ences among them in properties related to climate. Remarkable is its abundance on Earth,
as only the part that is in turbulent motion amounts to 1.34 × 109 Gt (not counting quanti-
ties that are stored in the soil, ground and glaciers), 260 times larger than the total mass of
0
2
4
6
8
10
12
14
16
18
20
180 200 220 240 260 280 300
Altitude (km)
Temperature (K)
With water vapour
Dry adiabatic
0
2
4
6
8
10
12
14
16
18
20
020 40 60 80 100 120 140 160 180 200 220 240
Altitude (km)
Net upward radiation flux (W/m²)
LW
LW –absorbed SW
Evapotranspiration
(latent heat)
50%
Conduction &
convection
(sensible heat)
11%
Net LW radiation
39%
Evapotranspiration
(latent heat)
34%
Conduction &
convection
(sensible heat)
7%
Net LW radiation
26%
SW radiation absorption
33%
34
the atmosphere. The turbulent motion of water, which is intrinsically uncertain, generates
climatic phenomena at all scales, from large-scale coupled ocean–atmosphere fluctua-
tions, such as the El Niño–Southern Oscillation (ENSO), Atlantic Multidecadal Oscillation
(AMO) and Interdecadal Pacific Oscillation (IPO), to regional droughts and floods. The
high specific heat (or heat capacity) of water, particularly in its liquid phase, combined with
its abundance, makes water the climatic thermostat of the Earth, i.e., the element that de-
termines the heat storage and through it the climate of the Earth. The high specific latent
heat of vaporization (calculated from Equation (6)) combined with the water occurrence
on Earth in all three phases, makes water the thermodynamic regulator of climate. Finally,
the fact that water is a universal solvent makes it an elixir of life, complementary to the CO₂
which is the other elixir of life, as through the photosynthesis it is responsible for the or-
ganic matter on which life on Earth is based.
Thus, the biosphere strongly depends on both CO₂ and H₂O. In particular, the pres-
ence of water determines the type and extent of ecosystems. In turn, the ecosystems affect
climate at large through the carbon and oxygen cycles (where the vast majority of the CO₂
and O₂ emissions are products of life, through respiration and photosynthesis, respec-
tively), and their contribution in the water cycle (transpiration) and in the energy cycle
(photosynthesis). Humans, as part of the biosphere, also interact with water and climate,
affecting them and being affected by them. Excluding human influences, the processes of
the biosphere determine the vast majority (96%) of CO₂ emissions and partly, in the ter-
restrial part, the emission of H₂O by the transpiration process. And as Koutsoyiannis et
al. [12,13] have shown, it is the relationship between temperature and biosphere that has
determined the recent increase in the atmospheric [CO₂].
Considering all these facts, it is stunning that the whole “climate project”, including
climate modelling, is based on hypotheses and scenarios about human CO₂ emissions.
8. Conclusions
According to the calculations presented here and the depiction of the results in Figure
24, the contribution of CO₂ to the greenhouse effect is 4% – 5%. Of this, 4% is due to human
emissions, which means that the total human contribution on the enhancement of the
greenhouse effect is 0.16% – 0.20% —a negligible effect. Irrespective of the origin of the
increase of [CO₂] in the last century, its contribution to the greenhouse effect is about 0.5%,
below any threshold to make it observable. In contrast, water (including clouds) contrib-
utes to the atmospheric greenhouse effect by 87% – 95%. In addition, 50% of Earth’s cool-
ing and atmosphere’s warming is due to water (against 39% due to LW radiation, which
again is dominated by water—Figure 26).
Common arguments trying to amplify the importance of human carbon emissions
are that these accumulate to the atmosphere and that they cause temperature increase.
The former argument is mistaken as the atmosphere does not have any mechanism to
separate the incoming CO₂ according to its origin, and to accumulate that part that comes
from humans. Also, the second argument has been refuted by showing, using modern
instrumental CO₂ and temperature time series, that temperature changes precede CO₂
changes and thus the CO₂ increase cannot be a cause of the temperature increase [11-
13,
78
].
Given these facts, the case of the magnified importance of CO₂, and particularly the
human emissions thereof, appears to be a historical accident in scientific terms, that was
exploited in non-scientific terms. If we return to science, the proper path is to improve
hydrology and stochastics in order to better understand climate. For climate is mostly hy-
drology in terms of its driving physical mechanisms (as articulated here) and mostly sto-
chastics in terms of its proper mathematical representation (as implied by its very defini-
tion; cf. Koutsoyiannis 2021, 2023).
Supplementary Materials: Not available.
Author Contributions: Not applicable.
Funding: This research received no funding but was conducted for scientific curiosity.
35
Data Availability Statement: This research uses no new data. The data sets used have been retrieved
from the sources described in detail in the text..
Acknowledgments: I thank an anonymous reviewer of a predecessor paper [18], whose critical com-
ments made it necessary to delve into the topic examined in the present paper. The brief reply to those
review comments was not included in the predecessor paper in order not to distract its focus. Yet it
constituted the springboard to produce this paper.
I particularly thank William Happer for his encouragement and his helpful comments on an earlier draft
of the paper.
Conflicts of Interest: The author declares no conflicts of interest.
Appendix A: MODTRAN locality profiles
MODTRAN implements five different locality profiles, which differ in their temper-
ature, H₂O and O₃ profiles. The profiles most relevant for this study are depicted Figure
A1. (Additional information can be found in [
79
].) By comparison with zonal temperature
distributions analogous to that of in Figure 4, but separately for summer and winter, we
infer that the tropical profile corresponds roughly to the equator, but is representative for
the entire torrid zone (between 23.4° N or S), the midlatitude profile corresponds to a lat-
itude at about 45° N or S, and the subarctic profile at about the latitude of polar circles
(66.6° N or S). In addition, MODTRAN includes the 1976 U.S. Standard Atmosphere tem-
perature profile, which provides an effective median for the set of locality profiles; this
was not used in this study.
Figure A2 depicts a characteristic example of MODTRAN output profiles upward,
downward and net LW radiation flux across the atmosphere for the tropical profile and
default settings.
Figure A1 Standard profiles of the indicated variables used in MODTRAN.
0
10
20
30
40
50
60
70
80
90
100
0.001 0.01 0.1 1 10 100 1000
Altitude (km)
Pressure (hPa)
Tropical
Midlatitude summer
Midlatitude winter
Subarctic summer
Subarctic winter
0
10
20
30
40
50
60
70
80
90
100
0% 20% 40% 60% 80% 100%
Altitud e (km)
Relative humidity
Tropical
Midlatitude summer
Midlatitude winter
Subarctic summer
Subarctic winter
0
10
20
30
40
50
60
70
80
90
100
180 200 220 240 260 280 300
Altitud e (km)
Temperature (K)
Tropical
Midlatitude summer
Midlatitude winter
Subarctic summer
Subarctic winter
0
10
20
30
40
50
60
70
80
90
100
0.00001 0.0001 0.001 0.01 0.1 1 10
Altitud e (km)
Water vapour pressure (hPa)
Tropical
Midlatitude summer
Midlatitude winter
Subarctic summer
Subarctic winter
36
Figure A2 Profiles of the indicated variables resulting from MODTRAN for the tropical profile and
default settings.
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