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Radiation Transfer Calculations and Assessment of Global Warming by CO 2

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  • Helmut Schmidt University Hamburg
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Radiation Transfer Calculations and Assessment of Global Warming by CO 2

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We present detailed line-by-line radiation transfer calculations, which were performed under different atmospheric conditions for the most important greenhouse gases water vapor, carbon dioxide, methane, and ozone. Particularly cloud effects, surface temperature variations, and humidity changes as well as molecular lineshape effects are investigated to examine their specific influence on some basic climatologic parameters like the radiative forcing, the long wave absorptivity, and back-radiation as a function of an increasing CO2 concentration in the atmosphere. These calculations are used to assess the CO2 global warming by means of an advanced two-layer climate model and to disclose some larger discrepancies in calculating the climate sensitivity. Including solar and cloud effects as well as all relevant feedback processes our simulations give an equilibrium climate sensitivity of Cs †= 0.7°C (temperature increase at doubled CO2) and a solar sensitivity of Ss† = 0.17°C (at 0.1% increase of the total solar irradiance). Then CO2 contributes 40% and the Sun 60% to global warming over the last century.
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Research Article
Radiation Transfer Calculations and Assessment of
Global Warming by CO2
Hermann Harde
Experimental Physics and Materials Science, Helmut-Schmidt-University, Holstenhofweg 85, 22043 Hamburg, Germany
Correspondence should be addressed to Hermann Harde; harde@hsu-hh.de
Received 29 June 2016; Revised 3 October 2016; Accepted 1 November 2016; Published 20 March 2017
Academic Editor: Bin Yu
Copyright ©  Hermann Harde. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present detailed line-by-line radiation transfer calculations, which were performed under dierent atmospheric conditions
for the most important greenhouse gases water vapor, carbon dioxide, methane, and ozone. Particularly cloud eects, surface
temperature variations, and humidity changes as well as molecular lineshape eects are investigated to examine their specic
inuence on some basic climatologic parameters like the radiative forcing, the long wave absorptivity, and back-radiation as a
function of an increasing CO2concentration in the atmosphere. ese calculations are used to assess the CO2global warming
by means of an advanced two-layer climate model and to disclose some larger discrepancies in calculating the climate sensitivity.
Including solar and cloud eects as well as all relevant feedback processes our simulations give an equilibrium climate sensitivity of
𝑆= .C(temperatureincreaseatdoubledCO
2) and a solar sensitivity of 𝑆= .C (at .% increase of the total solar irradiance).
en CO2contributes % and the Sun % to global warming over the last century.
1. Introduction
e Fih Assessment Report (AR) [] of the Intergov-
ernmental Panel on Climate Change (IPCC), a list of all
abbreviations is found in the annex, announces new evidence
of an anthropogenic climate change based on many inde-
pendent scientic analyses from observations of the climate
system, paleoclimate archives, theoretical studies of climate
processes, and simulations using climate models. So, the
IPCC classies the human inuence as extremely likely to be
thedominantcauseoftheobservedwarmingsincethemid-
th century (AR-WG-SPM-D). Increasing emissions of
carbon dioxide (CO2) over the last century especially are
made responsible for this change, and the equilibrium cli-
mate sensitivity (ECS or 𝑆) as a measure for the Earth’s
temperature increase at doubled CO2concentration in the
atmosphere is specied to be likely in the range .Cto.
C
(high condence) (AR-WG-SPM, p).
Although in all these elds of climate sciences great
progress has been achieved over the last decades and our
knowledge about the Earth-atmosphere system (EASy) could
signicantly be improved, explanations of the observed
global warming over the last century in particular the anth-
ropogenic contributions to this warming are still quite con-
tradictorily discussed.
All the more it is surprising
(i) that many of the consulted analyses and also the AR
itself do not better and clearly distinguish between
an anthropogenic emission of CO2and a naturally
generated part, where the latter even contributes more
than % to the overall emission, and its generation
rate and the respective absorption rate sensitively
respondonglobaltemperaturevariations;
(ii) that the IPCC claims it would be extremely likely
that more than half of the observed increase in global
average surface temperature from  to  was
caused by the anthropogenic increase in greenhouse
gas concentrations and other anthropogenic forcings,
while contributions from natural forcing and an
internal variability both would only likely be in the
range of .Cto.
C;
(iii) that the meanwhile well known delayed response of
CO2and methane (CH4)toseaandairtemperature
changes (see, e.g., Petit et al. []; Monnin et al. [];
Hindawi
International Journal of Atmospheric Sciences
Volume 2017, Article ID 9251034, 30 pages
https://doi.org/10.1155/2017/9251034
International Journal of Atmospheric Sciences
Caillon et al. []; Torn and Harte []; Humlum et
al. []; Salby []) are not considered in AR, and
respective consequences for an anthropogenic global
warming are not discussed;
(iv) that quite uncertain data about cloud feedbacks and
studies of the radiative forcing (RF) of greenhouse
(GH) gases are referred, which are mostly valid for
clear sky conditions, while the introduction of clouds
is usually omitted (AR-WG- Chap...);
(v) that the IPCC denies any noticeable solar inuence
on the actual climate, although strong evidence of an
increasing solar activity over the last century exists
(see, e.g., Hoyt & Schatten []; Willson & Mordvinov
[]; Shapiro et al. []; Ziskin & Shaviv []; Scafetta
& Willson []; Usoskin et al. []; Zhao & Feng [];
Soon et al. []);
(vi) that obviously important eects like convection and
evaporation feedback, which can contribute to sig-
nicant negative feedback (Harde- []), are not
considered in many analyses.
Nevertheless, despite these decits and simplications the
mean equilibrium climate sensitivity is specied with high
condence, and the GH gases are even assigned with very
high condence (%) to be responsible for the actual climate
changes.
Here we will focus on the assessment of one of the most
important quantities in climate sciences and its validation,
the ECS, which has to be scrutinized in more detail. Due to
its far reaching consequences for future climate predictions it
is particularly important to understand and to discover the
large discrepancies between dierent accounting methods
applied for this quantity. Also the weighting of some quite
dierent and even counteracting processes which control
ourclimate,butwhicharenotalwayswellunderstood,
has carefully to be investigated with its implications on the
climatesensitivity.Aquitecriticalreportofactuallypublished
ECS values and accounting methods expanded in AR has
been published by Lewis and Crok [].
In this contribution we will also retrace the main steps
of the IPCC’s preferred accounting system and compare this
with our own advanced two-layer climate model (LCM),
whichisespeciallyappropriatetocalculatetheinuenceof
increasing CO2concentrations on global warming as well
as the impact of solar variations on the climate (Harde-
 []). is model describes the atmosphere and the
ground as two layers acting simultaneously as absorbers and
Planck radiators, and it includes additional heat transfer
between these layers due to convection and evaporation. It
also considers short wave (sw) and long wave (lw) scattering
processes at the atmosphere and at clouds; it includes all
common feedback processes like water vapor, lapse rate,
and albedo feedback but additionally takes into account
temperature dependent sensible and latent heat uxes as
well as temperature induced and solar induced cloud cover
feedback.
e objective of our studies is not to present a new only
“true ECS” but to identify some of the dierent assumptions
and approximations with their far reaching consequences
in climate politics. It is without any doubt that the ECS
is the most important measure for the CO2inuence on
ourclimate,butitisalsoclearthatthisquantitydoesnot
distinguish between anthropogenic and natural CO2emis-
sions. erefore, as long as any natural variations in the CO2
concentrations are not accurately known, the ECS cannot be
used as a reliable indicator only for an anthropogenic global
warming. All this in mind the reader may have his own
reservations about the published data for this measure and
its signicance for a man-made climate change.
For the assessment of the ECS the IPCC favors the
concept of radiative forcing (RF), which is supposed to
be appropriate to describe the transition of the surface-
troposphere system from one equilibrium state to another
in response to an externally imposed perturbation. ere-
fore, in Section  we briey outline some basic relations
characterizing this concept, before we present in Section
detailed line-by-line radiation transfer (LBL-RT) calculations
for the lw up- and downwelling uxes in the atmosphere (for
details see Harde [, , ]), this for clear sky, global mean
cloud cover, full cloudiness, dierent surface temperatures,
humidity, and even dierent lineshapes. ese calculations
were performed for the most important GH gases water vapor
(WV), carbon dioxide, methane, and ozone and are based
ontheHITRANdatabase[].Sincetheconcentration
of the GH gases and the atmospheric pressure are changing
with temperature and altitude, the atmosphere has to be
segmented into up to  sublayers from ground to  km
height and in some cases additionally into three climate
zones. When these computations are repeated for dierent
CO2concentrations at otherwise same conditions, from the
changing uxes on the one hand the CO2initiated RF as
the main parameter for the IPCC accounting scheme and on
the other hand the sw and lw absorptivities as well as the
back-radiated fraction of the atmospheric emission as the key
parameters in our LCM can be derived.
Section  summarizes the main features of the LCM and
its calibration to satellite data, for the radiation and heat
uxes using the well known radiation and energy budget
scheme of Trenberth et al. [], for temperature changes due
to cloud cover variations applying the observations within the
International Satellite Cloud Climatology Project (ISCCP)
[]. With the respective key parameters of Section  inte-
grated in the climate model we simulate the Earth’s surface
temperature and the lower tropospheric temperature as a
function of the CO2concentration. e temperature increase
at doubled CO2concentration then directly gives the CO2
climate sensitivity. Such simulations reproduce the direct or
basic ECS value (without feedback processes), as specied
by the IPCC, within better than %. Signicant dierences,
however, can be observed with the dierent feedback eects
included. Our investigations show the largest discrepancies
for the WV and cloud feedback, but they also disclose the
importance of one of the primary stabilizers of the whole
climate system, the evaporation feedback. erefore, these
processes and their contributions are extensively analyzed
under dierent humidity, surface temperature, lapse rate, and
cloud cover conditions.
International Journal of Atmospheric Sciences
In particular, these studies show that the observed cloud
changes within the ISCCP cannot exclusively be explained by
pure thermally induced cloud cover changes but obviously are
additionally controlled by a further cloud forcing mechanism.
Since there exists strong evidence that the solar activity
alsohasapowerfulinuenceonthecloudcover,itis
reasonable to postulate such a solar induced cloud feedback
(see, e.g., Svensmark []; Haigh []). is is investigated
in detail in Section , dierentiating between a pure thermal
impact of an increasing solar activity and a nonthermally
inducedsolarcloudfeedback.Animportantcriterionfor
a validation, which mechanism might control such cloud
changes, can be derived from model simulations, which
includethesolaranomalyoverthelastcenturyandcompare
this directly with the observed global warming over this
period. ese investigations indicate that due to the strong
cloud feedback the observed warming over the last century
can only satisfactorily be explained, attributing a signicant
fractiontotheincreasedsolaractivityoverthisperiod(see
also Ziskin & Shaviv []; Vahrenholt & L¨
uning []).
Our simulations predict a solar contribution of about %
and a CO2induced contribution of % to global warming
over the last century with an equilibrium climate sensitivity of
.C, which is almost a factor of ve smaller than published
in AR.
2. Radiative Forcing
e concept of RF is well established in climate sciences and
used to assess the global warming as a result of an external
perturbation of EASy (AR-WG-Chap.). is perturbation
can result from solar anomalies, from increased GH gas
concentrations or volcanic activities. In all cases a direct
proportionality of the Earth’s temperature increase 𝐸as
response to an external forcing isassumedintheform
𝐸=𝑆⋅, ()
with 𝑆as the climate sensitivity parameter, also known as
Planck sensitivity. e negative reciprocal of 𝑆is called the
Planck feedback (see AR-WG-Chap.), although this is not
afeedbackprocessinthecommonsenselikethewatervapor,
the albedo, or lapse rate feedback, which all describe an
internal amplication or attenuation of an external pertur-
bation. e Planck feedback rather represents the change of
the outgoing long wave radiation (OLR), or more precisely
the change of the total upwelling lw intensity up
total with the
ground temperature. Since an increasing amount of OLR with
the temperature causes a negative feedback for EASy, it can be
dened as the negative partial derivative as follows:
OLR
𝐸=−up
total
𝐸=1
𝑆.(a)
Its value, deduced as an average from dierent climate
models, is given in AR-WG-Tab.. as . W/m2/C, and,
thus, the Planck sensitivity becomes 𝑆=.W
−1 m2∘
C. A
somewhat simplied method to assess the Planck feedback
and sensitivity, which agrees within % with these values, is
to use the slope of the Stefan-Boltzmann law at an eective
Earth temperature e =  K and emissivity e =1with
up
total
𝐴
𝑇e =255 K=−4
e3
e =−4up
total
e ⇒ 1
𝑆(b)
and to assume the same temperature response for the surface
as for the atmosphere. is results in a Planck feedback
of . W/m2/C and a climate sensitivity parameter 𝑆
=.W
−1 m2∘
C. is the Stefan-Boltzmann constant with
.−8 W/m2/C4and up
total the OLR with . W/m2.
Eq. () is the basis for many climate models (see AR
[] and AR []) and even for more complex mod-
els like the Atmosphere-Ocean-General-Circulation Models
(AOGCMs) this fundamental relation is used to express
theinuenceofthegrowingGHgasconcentrationsonthe
surfacetemperature.So,forthesemodelsandalsoforthe
observationally based energy budget ECS estimates (see,
e.g., Forster & Gregory []; Lindzen & Choi []) 
is a quite fundamental quantity, which cannot directly be
measured but has to be deduced from extensive spectroscopic
calculations or solar data, which on their part rely on many
individual observations and measurements. For quite actual
investigations of see also Feldman et al. [].
In this context it should be noticed that alternative
denitionsofRFhavebeendeveloped,eachwithitsown
advantages and limitations (see AR-WG-Chap.). Here
we only consider the instantaneous RF, which refers to an
instantaneous change in the net (down minus up) radiative
ux (sw plus lw) due to an imposed change. is forcing is
usually dened in terms of ux changes at the top of the
atmosphere (TOA) or at the tropopause.
In this contribution, particularly the inuence of CO2
on global warming is of interest. erefore, in the next
subsection we present some actual radiation transfer (RT)
calculations,fromwhichtheinstantaneousRFduetoincreas-
ing CO2in the atmosphere and also some related quantities,
which are of relevance for our two-layer climate model,
can be derived. Since for these model calculations it is not
sucient only to consider the net radiative uxes at the
tropopause, neglecting the downwelling absorption changes
in the troposphere and the upwelling absorption changes over
the stratosphere, we apply the RT concept from the surface to
TOA and vice versa. e sw absorption changes over the full
atmosphereandthisasafunctionoftheCO
2concentration
can be captured from our previous investigations (Harde-
 []).
As already outlined, an important reference for the
inuence of CO2is the temperature increase at doubled CO2
concentration under steady state conditions, which is known
astheequilibriumclimatesensitivityECSor𝑆,andthe
respective lw forcing may be designated as CO2.Sinceat
TOA a lw downwelling ux is zero, the forcing at TOA can be
dened as dierence of the total outgoing intensities up
total at
single and doubled CO2concentration :
CO2=up
total ()−up
total (2),()
International Journal of Atmospheric Sciences
where here we primarily consider the lw radiative forcing.
up
total can be explained to consist of the nonabsorbed terrestrial
intensity (𝐸−
abs) plus the emitted intensity of the atmo-
sphere in upward direction up
𝐴; thus, it holds the following:
CO2=
𝐸−
abs ()+up
𝐴()−
𝐸+
abs (2)
−up
𝐴(2)
=up
𝐴−
abs
()
with
up
𝐴=up
𝐴()−up
𝐴(2),
abs =
abs ()−
abs (2).()
𝐸is the intensity of the incident terrestrial radiation and abs
theabsorbeduxintheatmosphere.
So, due to () this forcing generally consists of two con-
tributions, the change in the upward atmospheric emission
and the change in the atmospheric absorption of terrestrial
radiation, both as response to a doubling of the CO2concen-
tration.
Expressing the downwelling atmospheric intensity down
𝐴
via the total emitted intensity total
𝐴and the fraction 𝐴,which
is directed downward (also called asymmetry factor of the
emitted atmospheric radiation) and which is generally found
from integrating the respective spectral intensities up,down
𝐴,̃
]
at TOA and the surface over the frequency or reciprocal
wavelength (wavenumber)
]with
𝐴=down
𝐴
total
𝐴=
0down
𝐴,̃
]
]
0down
𝐴,̃
]
]+∫
0up
𝐴,̃
]
],()
the dierence down
𝐴between single and doubled concentra-
tion gives
down
𝐴=down
𝐴()−down
𝐴(2)
=total
𝐴()⋅
𝐴()−total
𝐴(2)⋅
𝐴(2)
=total
𝐴()⋅
𝐴()−total
𝐴()−total
𝐴
⋅
𝐴()−
𝐴
=total
𝐴()⋅
𝐴+total
𝐴⋅
𝐴()−total
𝐴
⋅
𝐴,
()
with total
𝐴and 𝐴as the respective dierences of total
𝐴and
𝐴at single and doubled CO2concentration.
en, with (), for the dierence of the upwelling atmo-
spheric intensities we can write
up
𝐴=total
𝐴()⋅1−
𝐴()−total
𝐴(2)
⋅1−
𝐴(2)
=total
𝐴()−total
𝐴(2)−total
𝐴()⋅
𝐴()
+total
𝐴(2)⋅
𝐴(2)
=total
𝐴−total
𝐴()⋅
𝐴−total
𝐴⋅
𝐴()
+total
𝐴⋅
𝐴
=total
𝐴⋅1−
𝐴()+
𝐴−total
𝐴()⋅
𝐴,
()
andtheradiativeforcingatdoubledCO
2concentration can
nally be expressed as follows:
CO2=total
𝐴⋅1−
𝐴()+
𝐴−total
𝐴()
⋅
𝐴−
abs.()
is CO2forcing can directly be derived from radiation
transfer calculations (Schwarzschild [, ]; Goody and
Yung []; Salby- []; Harde- []; Harde- []),
by which the up- and downwelling uxes as well as the
absorption and emission in the atmosphere are computed.
3. Radiation Transfer Calculations
Since it is obvious that the cloud cover has a strong inuence
ontheup-anddownwellinguxesintheatmosphereandalso
on the strength of the GH eect, we have performed line-by-
line radiation transfer (LBL-RT) calculations under dierent
cloudiness conditions, ground temperatures, and humidity to
evaluatetheinuenceofCO
2on global warming. We also
briey investigate the inuence of lineshape eects on RT
calculations and their consequences for the ECS.
Here we only present global RT calculation with averaged
values for the temperature, water vapor concentration, and
an average lapse rate, since separate computations for the
tropics mid- and high-latitudes with individual proles and
averaging over the climate zones with an area weighting factor
gave almost the same results (see also Harde []).
3.1. Clear Sky at Global Mean Temperature. Ta b l e  s h o w s t h e
results of our LBL calculations under clear sky conditions
as a function of the CO2concentration, using a ground
temperature of 𝐸=.K=
Candaterrestrialintensity
of 𝐸=W/m
2.esevaluesareconsistentwiththoseused
by Trenberth et al. [].
Asanaverageoverthethreeclimatezonesthewatervapor
concentration at ground was assumed to be ,.ppm
and decreasing with altitude due to the Clausius-Clapeyron
equation (for details see Harde- []). ese data have
been derived from satellite measurements (Vey []) and
are almost a factor of two larger than those given by the
International Journal of Atmospheric Sciences
T : RT-calculations for dierent CO2concentrations at clear sky (𝐸=.W/m
2,𝐸= . K, water vapor at ground: .ppm,
CH4:.ppm,andO
3varying over the stratosphere with a maximum concentration of  ppm).
CO2
ppm up
total
W/m2up
𝐴
W/m2down
𝐴
W/m2total
𝐴
W/m2abs
W/m2aLW
%𝐴
%
. . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
– . . −4.46 −4.41 −5.00 −1.27 −0.35
up
𝐴−
abs .
US Standard Atmosphere []. But they are in quite good
agreement with the Average Global Atmosphere (Ollila []),
which even species values % larger than ours. e CH4
concentration was set to . ppm and O3to vary over the
stratosphere with a maximum concentration of  ppm at
kmaltitude.elapserateandpressurevariationwith
altitude agree with the US Standard Atmosphere.
Our calculations cover a spectral interval from  to
 cm−1, corresponding to .% of a Planck radiator at
𝐸, and they include more than , lines with spectral
intensities larger −24 cm−1/(moleculescm−2). e spectral
resolution for calculations at CO2concentrations of 
and  ppm, which are used as reference data, was chosen
as . cm−1 (. GHz), and for the vertical direction 
sublayers were distinguished, whereas the calculations at the
other concentrations were performed with half the spectral
and vertical resolution.
Comparison of the two last values in column  of Table 
shows that within rounding errors () is well conrmed.
Looking at the upwelling intensity up
𝐴oftheatmosphere(col-
umn ) we see that this is almost identical at single and dou-
bled CO2concentration, while the downwelling part down
𝐴
(column)andthetotalatmosphericradiationtotal
𝐴(column
) both are increasing by more than . W/m2.ismeans
that at clear sky conditions most of the additionally absorbed
terrestrial radiation at doubled CO2(see column ) is instan-
taneously directed back to the surface, and only a relatively
small contribution of . W/m2istrappedintheatmosphere.
Sinceinthiscasethersttwotermsin()almostexactly
compensate each other, the forcing under clear sky conditions
is in good approximation only determined by the absorption
dierence. us, from the simple radiative forcing model ()
it follows that also the ground temperature variations are
almost only determined by this additionally absorbed power.
In general, however, it should be clear that the surface
temperature also depends on the emission characteristic of
the atmosphere, which changes with the GH gas concentra-
f
Aclear sky
log t f
A
aLW clear sky
log t aLW
84
82
80
78
76
74
72
64
63
62
61
60
59
58
CO2-concentration (ppm)
f
A(%)
8007006005004003002001000
aLW (%)
F : Lw absorptivity aLW (black diamonds) and downwelling
fraction 𝐴of emitted atmospheric radiation (blue triangles) at clear
sky with respective logarithmic approximations.
tion as well as with the cloud cover and is determined by the
rst two terms in (). Only under clear sky they just feed most
of the additionally absorbed power back to the surface.
Inamoreadvancedmodel,asthiswillbediscussedlater
with a radiation and energy balance at TOA and at the surface,
including heat uxes of sensible and latent heat or from
neighbouringclimatezones,aswellasswandlwabsorption
losses, even at clear sky conditions the atmospheric emission
characteristic is of relevance, which can well be represented
by the fraction 𝐴ofthedownwardemittedradiation.
e relative absorption of terrestrial radiation by the GH
gases and its variation with increasing CO2concentration are
listed in column . It is normalized to the incident terrestrial
ux and in this way represents the lw absorptivity aLW as
dened in Harde- [] (4).isabsorptivityisrepre-
sented in Figure  (black diamonds) together with the back-
radiated fraction 𝐴of the atmosphere (blue triangles). At
higher concentrations both quantities can well be represented
International Journal of Atmospheric Sciences
T : RT-calculations for CO2concentrations of  and  ppm at a surface temperature of .C (. K) and clear sky (𝐸=
 W/m2, water vapor at ground: ,. ppm, CH4of . ppm and O3varying over the stratosphere with a maximum concentration of
ppm).
CO2
ppm up
total
W/m2up
𝐴
W/m2down
𝐴
W/m2total
𝐴
W/m2abs
W/m2aLW
%𝐴
%
 . . . . . . .
 . . . . . . .
. . −4.32 −3.64 −4.78 −1.14 −0.38
by logarithmic graphs (magenta line for aLW ,greenlinefor
𝐴), indicating that due to saturation eects then only the far
wings will further contribute to an increasing absorption or
emission.
3.2. RT Calculations at Clear Sky and Increased Surface Tem-
perature. Whereastheabovecalculationsreectasomewhat
articial situation, assuming clear sky and a global mean
temperature of C, we know that with declining cloud cover
the average surface temperature is signicantly climbing up.
So, from observations within the International Satellite Cloud
Climatology Project (ISCCP) [] it is found that around
the global mean cloud cover of % a reducing cloudiness
of % causes a temperature increase of about .–.C, a
tendencywhichisalsoconrmedbydataofHartmann[].
erefore, to get a better understanding of how such
higher temperature modies the respective uxes in the
atmosphere and especially this aects the CO2radiative
forcing, we have performed additional calculations for a
surface temperature of .C. At this temperature and a
surface emissivity of % the terrestrial radiation climbs
up to  W/m2. But with the higher temperature also the
humidity increases and therefore aects the absorptivity of
the GH gases. From the measured water vapor concentrations
as well as the temperatures at mid-latitudes and the tropics we
deduce at the higher temperature a growth of the vapor con-
centration of , ppm with a new total mean concentration
near the surface of , ppm.
Calculations for this higher ground temperature and
water vapor concentration are shown in Table .
Since for an evaluation of CO2with its inuence on the
radiation budget it is sucient to concentrate on the uxes at
single and doubled CO2concentration, here we only present
this reduced data set. Compared to the lower temperature
calculations the radiative forcing increases by . W/m2,
which is mainly due to changes in the upward atmospheric
emission with up
𝐴=.W/m
2, while the absorption changes
with abs =. W/m2are even getting slightly smaller. Also
thedownwarddirecteduxchangewithdown
𝐴=. W/m2
and the total atmospheric emission change with total
𝐴=
. W/m2are reducing, although the absolute uxes and
the absorption are considerably larger, caused by the higher
temperature and water vapor.
e absorptivity as calculated under these modied
conditions is shown in Figure  (black diamonds) together
with the back-radiated fraction 𝐴of the atmosphere (blue
triangles). At higher CO2concentrations both graphs can
f
Aclear sky
log t f
A
aLW clear sky
log t aLW
84
86
82
80
78
76
74
72
64
63
62
61
60
59
58
CO2-concentration (ppm)
f
A(%)
8007006005004003002001000
aLW (%)
F : Lw absorptivity aLW (black diamonds) and downwelling
fraction 𝐴of atmospheric radiation at 𝐸= .Candclearsky
with respective logarithmic approximations.
again well be approximated by logarithmic functions, similar
to Figure  (magenta line for aLW , green line for 𝐴), only on
slightly shied scales.
3.3. LBL-RT Calculations at Global Mean Cloud Cover. Under
regular cloud cover signicant changes from clear sky condi-
tionshavetobeexpectedfortheup-anddownwellinguxes
due to the strong shielding eect of clouds not only for sw but
also for lw radiation. In our radiation transfer calculations
the clouds are considered as a single thinner layer, which
absorbs the incident lw radiation from up or down direction
with a cloud absorptivity aLC and which has a transmissivity
of 1−aLC, while lw scattering processes at the clouds are
here neglected. For the RT calculation only the resulting
attenuation of radiation, which passes the layer, is of rele-
vance, and for these calculations it is sucient to express the
total attenuation uniformly only by one parameter, the cloud
absorptivity. In addition to the transmitted radiation the
clouds are also emitting as a Planckian radiator with an emis-
sivity 𝜀LC and a temperature determined by the cloud altitude.
e Global Energy and Water Cycle Experiment
(GEWEX) Radiation Panel, which compares the available
global long-term cloud data products with the ISCCP
and consists of  satellite measurement teams, species
the global total cloud amount between % and %
(Stubenrauch et al. []), where this relatively wide range
is mainly explained by dierent instrument sensitivity and
by retrieval methodology. For our further calculations we
assume a global mean cloud cover of 𝐶= % in agreement
International Journal of Atmospheric Sciences
T : RT-calculations for dierent CO2concentrations at mean cloud cover of %, a cloud altitude of . km, and cloud emissivity of %
(𝐸=W/m
2,𝐸= . K = .C, water vapor at ground of ,. ppm, CH4of . ppm, and O3varying over the stratosphere with a
maximum concentration of  ppm).
CO2
ppm up
total
W/m2up
𝐴
W/m2down
𝐴
W/m2total
𝐴
W/m2abs
W/m2aLW
%𝐴
%
. . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
 . . . . . . .
. . −3.45 −2.61 −2.93 −0.74 −0.35
up
𝐴−abs .
with the ISCCP []. e cloud altitude with 𝐶=.km
and the emissivity with 𝜀LC =%werechosentoalmost
exactly reproduce the widely accepted radiation and energy
budget scheme of Trenberth et al. [] (hereaer called
TFK-scheme) with a total upward ux of up
total = 239.2W/m2
and a nonabsorbed outgoing radiation from the surface with
up
total −up
𝐴=40W/m2.
Our RT calculations under otherwise identical conditions
as described in Section . and at a surface temperature 𝐸=
C are listed in Table . ey show signicant changes to
the radiation uxes under clear sky, for example, for down
𝐴,
more than  W/m2or for total
𝐴even  W/m2.Alsothe
total absorption, now additionally increased by the clouds, is
 W/m2larger than at clear sky.
But respective dierences between single and doubled
CO2concentrations, which are here of main interest, in
general are getting smaller. So, the radiative forcing reduces
to CO2= . W/m2andisalmostidenticalwiththe
gure used by the IPCC (Myhre et al. []), although here we
consider only the instantaneous lw RF at TOA. is forcing
istheresultoftheupwellingatmosphericcontributionof
up
𝐴= 0.84W/m2andontheotherhandareduced
absorption dierence of abs = −2.93W/m2compared to
W/m
2at clear sky. Also changes in the back-radiation
diminish from . W/m2to down
𝐴= −3.5W/m2and the
total emission even from . W/m2to total
𝐴=−2.6W/m2.
Because of this smaller alteration in the atmospheric
emission the st term in () with total
𝐴⋅(1−
𝐴()+𝐴)
now reduces to . W/m2,whereasthendtermwith
−total
𝐴()⋅𝐴=.W/m
2dominates. From the total forcing
of . W/m2in this case . W/m2goes back to the surface,
while . W/m2remains in the atmosphere.
To deduce the lw absorptivity at cloudiness from the radi-
ation transfer calculations, the absorbed intensity listed in
f
A66% clouds
log t f
A
aLW 66% clouds
log t aLW
83
81
79
77
75
63
62
61
60
59
CO2-concentration (ppm)
f
A(%)
8007006005004003002001000
aLW (%)
F : Lw absorptivity aLW (black diamonds) and downwelling
fraction 𝐴of emitted atmospheric radiation at % cloud cover
with respective logarithmic approximations.
column  of Table , which now reects the total absorption
caused by the GH gases and the clouds, has to be reduced
by the cloud fraction. Such correction seems appropriate for
a direct comparison of the absorptivity changes caused by
the GH gases at clear sky and clouds, and it is necessary for
our two-layer climate model, where the cloud absorption is
already integrated in the model and represented by respective
parameters. Such a correction is done by comparing the
absorption, for example, at  ppm with and without clouds
at the same ground temperature, and reducing column  by
this dierence. Normalizing the corrected absorption to the
incident radiation then gives in good approximation the lw
absorptivity aLW oftheGHgasesinthepresenceofclouds.
FigureshowstherespectiveabsorptivityaLW (black
diamonds) and the back-radiated fraction 𝐴(blue triangles)
at a cloud cover of %. Again both quantities can well be
International Journal of Atmospheric Sciences
T : RT-calculations with MRT lineshape for CO2concentrations of  and  ppm at mean cloud cover of %, a cloud altitude of
. km, and cloud emissivity of .% (𝐸=W/m
2,𝐸= . K = .C, water vapor at ground of ,. ppm, CH4of . ppm, and O3
varying over the stratosphere with a maximum concentration of  ppm).
CO2
ppm up
total
W/m2up
𝐴
W/m2down
𝐴
W/m2total
𝐴
W/m2abs
W/m2aLW
%𝐴
%
 . . . . . . .
 . . . . . . .
. . −3.18 −2.20 −2.82 −0.71 −0.34
represented by logarithmic graphs (magenta line for aLW ,
greenlinefor𝐴). While 𝐴slightly increases by .%,
𝐴remains constant. However, the absorptivity changes
almost decline by a factor of two to .% at doubled CO2
concentration.
Simulations at higher cloud altitudes under otherwise
same constrains give a reduced upwelling ux up
total and a
further diminishing radiative forcing CO2.
3.4. RT Calculations with MRT Lineshape. While the pre-
cedingcalculationsarebasedonthestandardmolecular
collision theory, considering collisional broadening of spec-
tral transitions, which are characterized by a Lorentzian
lineshapeorathigheraltitudesalsobyaVoigtprole,we
have also performed extensive calculations using a more
sophisticatedlineshapeasgivenbythemolecularresponse
theory (MRT) (Harde et al. [–]). is theory represents
a generalization and unication of the classical collision
theories of Lorentz and on the other hand of van Vleck and
Weisskopf, considering thermalization of molecules during a
collision, which in its limit is determined by the reciprocal of
the molecular transition frequency and controlled by Heisen-
berg’s uncertainty principle (Harde & Grischkowsky []).
e specialty applying MRT is that it not only describes the
near resonance absorption and emission behavior as already
adequately characterized by a Lorentzian, but also reects the
far wing response, which is mainly caused by the ultrafast
time response of molecules to an external electric eld during
collisions. So, the MRT lineshapes already represent very
well the continuum water vapor absorption extending up
to  cm−1 and caused by the low frequency water vapor
absorption band around  cm−1 (see Burch & Gryvnak
[]; Roberts et al. []).
is continuum water vapor background together with
the far wing contributions of the other molecules signicantly
modies the absolute up- and downwelling uxes. erefore,
tofurthersatisfytheradiationandenergybalanceoftheTFK-
scheme, in the radiation transfer calculations the cloud alti-
tude has to be increased to  km and the cloud absorptivity =
emissivity reduced to .%.
e results of such a calculation at a mean cloud cover
of % and at otherwise same conditions as described in
Section . is shown in Table . To compare the alterations
caused by doubling the CO2concentration it is again su-
cient only to display the calculations for  and  ppm
CO2. It is interesting to see that the more sophisticated
lineshape and the continuum background absorption almost
have no inuence on CO2, the RF at doubled CO2,which
agrees within better than % with the simpler calculations
shown in Table , neglecting the continuum absorption and
using a Lorentzian lineshape. Also changes in the other
uxes at doubled CO2concentration are consistent within
. W/m2; only the back-radiation changes to down
𝐴=
−3.18W/m2and is about . W/m2smaller; thus, the power
le in the atmosphere increases by this amount.
For this insensitivity to a dierent collision and line-
shapetheoryweseethreemainreasons:rst,becauseof
the larger absolute ux variations, caused by the dierent
wing absorption and WV continuum, the outgoing uxes
were recalibrated to the TFK-scheme via cloud altitude
andabsorptivitytoensurecomparableabsoluteuxesin
agreement with the observations. Second, we only consider
ux dierences at single and doubled CO2concentration for
the same shape, so that discrepancies to another shape are
not directly observed and smaller absolute variations with the
CO2concentration are to some extent cancel out. ird, the
interference of the CO2spectrum with strong water vapor
lines and the underlying WV continuum further attenuates
any CO2lineshape eects.
So, calculations based on the classical collision theory
obviously reproduce quite reliable data of any radiation
changes in the atmosphere. Since the far wings of the CO2
lines are found to decay more rapidly than a Lorentzian and
thus should contribute less to the total absorption (see, e.g.,
Edwards and Strow []), such calculations even simulate
slightly worse conditions and result in a more conservative
assessment of global warming by CO2.ForactualCO
2line-
shape studies see also Happer [], and for RT calculations
including CO2line mixing see Mlynczak et al. [], and Ozak
et al. [].
3.5. RT Calculations for Total Cloud Cover. With a further
increasing cloud cover and thus a stronger shielding of the
sw radiation also the surface temperature further drops, and
owing to the ISCCP observations at % cloudiness then
a reduced temperature of approximately .Ccomparedto
mean cloudiness is expected.
Respective radiation transfer calculations at % cloud
cover, a surface temperature of .Cwithaterrestrial
intensity of  W/m2, and a further reduced water vapor
concentration at ground of , ppm, otherwise using the
same conditions as in Section ., are shown in Table .
e total outgoing radiation up
total drops by . W/m2and
the total atmospheric emission total
𝐴by . W/m2compared
International Journal of Atmospheric Sciences
T : RT-calculations for dierent CO2concentrations at % cloudiness, a cloud altitude of .km, and cloud emissivity of %
(𝐸=W/m
2,𝐸= . K = .C, water vapor: ,. ppm, CH4:.ppm,andO
3varying over the stratosphere with a maximum
concentration of  ppm).
CO2
ppm up
total
W/m2up
𝐴
W/m2down
𝐴
W/m2total
𝐴
W/m2abs
W/m2aLW
%𝐴
%
 . . . . . . .
 . . . . . . .
. . −2.93 −1.90 −1.93 −0.50 −0.34
f
A100% clouds
log t f
A
aLW 100% clouds
log t aLW
82
81
80
79
77
78
63
64
62
61
60
59
CO2-concentration (ppm)
f
A(%)
8007006005004003002001000
aLW (%)
F : Lw absorptivity aLW (black diamonds) and downwelling
fraction 𝐴(blue triangles) of emitted atmospheric radiation at
% cloudiness and a ground temperature of .Cwithrespective
logarithmic approximations.
to a cloud cover of %. Nevertheless, the forcing due to
doubled CO2reduces to CO2= . W/m2,consistingof
a slightly increasing contribution up
𝐴= 1.0W/m2and a
decreasing absorption abs = −1.9 W/m2.Alsotheabsolute
downwelling radiation down
𝐴further decreases by . W/m2,
and its alteration at doubled CO2concentration reduces to
. W/m2, so almost nothing remains in the atmosphere.
e absorptivity aLW as listed in column  is again
found by comparing the absorption at  ppm with clouds
(. W/m2) and without clouds (. W/m2), reducing
column  by this dierence and then normalizing the cor-
rected absorption to the incident radiation. aLW is shown
together with the parameter 𝐴in Figure . Both graphs can
againwellberepresentedbylogarithmicplots.
e changes in absorptivity from single to doubled
CO2with aLW =.% further decline compared to %
cloudiness or clear sky, and also 𝐴is slightly reducing to
.%.
So,fromthesecalculationsitisclearthattheup-and
downwelling uxes and also the absorption in the atmosphere
are signicantly varying with the cloud clover and, therefore,
they will also signicantly inuence the global mean surface
temperature. But it can also well be recognized, that changes
with the CO2concentration by trend get smaller with increas-
ing cloudiness.
4. Assessment of CO2Global Warming
Dierent to general circulation models, which try to predict
local climate variations over some time period and, there-
fore, have to solve complex coupled nonlinear dierential
equations with a large number of parameters, making these
calculations extremely time